Theorem tfrlem11 3906

Description: Lemma for transfinite recursion. Compute the value of C.

Hypotheses

Ref	Expression
tfrlem.1	|- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
tfrlem.2	|- F = U.A
tfrlem.3	|- C = (F u. {<.dom F, (G` (F |` dom F))>.})

Assertion

Ref	Expression
tfrlem11	|- (dom F e. On -> (y e. suc dom F -> (C` y) = (G` (C |` y))))

Distinct variable groups:   x,y,f,A   x,F,y,f   x,G,y,f   x,C,y,f

Proof of Theorem tfrlem11

Step	Hyp	Ref	Expression
1		ssun1 2183	. . . . . . . . 9 |- F (_ (F u. {<.dom F, (G` (F |` dom F))>.})
2		tfrlem.3	. . . . . . . . 9 |- C = (F u. {<.dom F, (G` (F |` dom F))>.})
3	1, 2	sseqtr4 2084	. . . . . . . 8 |- F (_ C
4		funssfv 3720	. . . . . . . . . . . 12 |- ((Fun C /\ F (_ C /\ y e. dom F) -> (C` y) = (F` y))
5	4	3expa 831	. . . . . . . . . . 11 |- (((Fun C /\ F (_ C) /\ y e. dom F) -> (C` y) = (F` y))
6	5	adantrl 394	. . . . . . . . . 10 |- (((Fun C /\ F (_ C) /\ (dom F e. On /\ y e. dom F)) -> (C` y) = (F` y))
7		fun2ssres 3539	. . . . . . . . . . . . 13 |- ((Fun C /\ F (_ C /\ y (_ dom F) -> (C |` y) = (F |` y))
8	7	3expa 831	. . . . . . . . . . . 12 |- (((Fun C /\ F (_ C) /\ y (_ dom F) -> (C |` y) = (F |` y))
9	8	fveq2d 3713	. . . . . . . . . . 11 |- (((Fun C /\ F (_ C) /\ y (_ dom F) -> (G` (C |` y)) = (G` (F |` y)))
10		onelsst 2990	. . . . . . . . . . . 12 |- (dom F e. On -> (y e. dom F -> y (_ dom F))
11	10	imp 350	. . . . . . . . . . 11 |- ((dom F e. On /\ y e. dom F) -> y (_ dom F)
12	9, 11	sylan2 451	. . . . . . . . . 10 |- (((Fun C /\ F (_ C) /\ (dom F e. On /\ y e. dom F)) -> (G` (C |` y)) = (G` (F |` y)))
13	6, 12	eqeq12d 1481	. . . . . . . . 9 |- (((Fun C /\ F (_ C) /\ (dom F e. On /\ y e. dom F)) -> ((C` y) = (G` (C |` y)) <-> (F` y) = (G` (F |` y))))
14		tfrlem.1	. . . . . . . . . 10 |- A = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
15		tfrlem.2	. . . . . . . . . 10 |- F = U.A
16	14, 15	tfrlem9 3904	. . . . . . . . 9 |- (y e. dom F -> (F` y) = (G` (F |` y)))
17	13, 16	syl5bir 210	. . . . . . . 8 |- (((Fun C /\ F (_ C) /\ (dom F e. On /\ y e. dom F)) -> (y e. dom F -> (C` y) = (G` (C |` y))))
18	3, 17	mpanl2 705	. . . . . . 7 |- ((Fun C /\ (dom F e. On /\ y e. dom F)) -> (y e. dom F -> (C` y) = (G` (C |` y))))
19	14, 15, 2	tfrlem10 3905	. . . . . . . 8 |- (dom F e. On -> C Fn suc dom F)
20		fnfun 3571	. . . . . . . 8 |- (C Fn suc dom F -> Fun C)
21	19, 20	syl 10	. . . . . . 7 |- (dom F e. On -> Fun C)
22	18, 21	sylan 448	. . . . . 6 |- ((dom F e. On /\ (dom F e. On /\ y e. dom F)) -> (y e. dom F -> (C` y) = (G` (C |` y))))
23	22	exp32 377	. . . . 5 |- (dom F e. On -> (dom F e. On -> (y e. dom F -> (y e. dom F -> (C` y) = (G` (C |` y))))))
24	23	pm2.43i 64	. . . 4 |- (dom F e. On -> (y e. dom F -> (y e. dom F -> (C` y) = (G` (C |` y)))))
25	24	pm2.43d 65	. . 3 |- (dom F e. On -> (y e. dom F -> (C` y) = (G` (C |` y))))
26		opeq1 2478	. . . . . . . . . . 11 |- (y = dom F -> <.y, (G` (C |` y))>. = <.dom F, (G` (C |` y))>.)
27	26	adantl 388	. . . . . . . . . 10 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. = <.dom F, (G` (C |` y))>.)
28	3, 7	mp3an2 901	. . . . . . . . . . . . 13 |- ((Fun C /\ y (_ dom F) -> (C |` y) = (F |` y))
29		eqimss 2099	. . . . . . . . . . . . 13 |- (y = dom F -> y (_ dom F)
30	28, 21, 29	syl2an 454	. . . . . . . . . . . 12 |- ((dom F e. On /\ y = dom F) -> (C |` y) = (F |` y))
31		reseq2 3353	. . . . . . . . . . . . 13 |- (y = dom F -> (F |` y) = (F |` dom F))
32	31	adantl 388	. . . . . . . . . . . 12 |- ((dom F e. On /\ y = dom F) -> (F |` y) = (F |` dom F))
33	30, 32	eqtrd 1499	. . . . . . . . . . 11 |- ((dom F e. On /\ y = dom F) -> (C |` y) = (F |` dom F))
34		fveq2 3709	. . . . . . . . . . 11 |- ((C |` y) = (F |` dom F) -> (G` (C |` y)) = (G` (F |` dom F)))
35		opeq2 2479	. . . . . . . . . . 11 |- ((G` (C |` y)) = (G` (F |` dom F)) -> <.dom F, (G` (C |` y))>. = <.dom F, (G` (F |` dom F))>.)
36	33, 34, 35	3syl 20	. . . . . . . . . 10 |- ((dom F e. On /\ y = dom F) -> <.dom F, (G` (C |` y))>. = <.dom F, (G` (F |` dom F))>.)
37	27, 36	eqtrd 1499	. . . . . . . . 9 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. = <.dom F, (G` (F |` dom F))>.)
38	37	sneqd 2409	. . . . . . . 8 |- ((dom F e. On /\ y = dom F) -> {<.y, (G` (C |` y))>.} = {<.dom F, (G` (F |` dom F))>.})
39		opex 2772	. . . . . . . . 9 |- <.y, (G` (C |` y))>. e. V
40	39	snid 2425	. . . . . . . 8 |- <.y, (G` (C |` y))>. e. {<.y, (G` (C |` y))>.}
41	38, 40	syl5eleq 1546	. . . . . . 7 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. e. {<.dom F, (G` (F |` dom F))>.})
42		elun2 2188	. . . . . . 7 |- (<.y, (G` (C |` y))>. e. {<.dom F, (G` (F |` dom F))>.} -> <.y, (G` (C |` y))>. e. (F u. {<.dom F, (G` (F |` dom F))>.}))
43	41, 42	syl 10	. . . . . 6 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. e. (F u. {<.dom F, (G` (F |` dom F))>.}))
44	43, 2	syl6eleqr 1551	. . . . 5 |- ((dom F e. On /\ y = dom F) -> <.y, (G` (C |` y))>. e. C)
45		fvex 3717	. . . . . . 7 |- (G` (C |` y)) e. V
46	45	fnopfvb 3739	. . . . . 6 |- ((C Fn suc dom F /\ y e. suc dom F) -> ((C` y) = (G` (C |` y)) <-> <.y, (G` (C |` y))>. e. C))
47		visset 1804	. . . . . . 7 |- y e. V
48	47	eqelsuc 3044	. . . . . 6 |- (y = dom F -> y e. suc dom F)
49	46, 19, 48	syl2an 454	. . . . 5 |- ((dom F e. On /\ y = dom F) -> ((C` y) = (G` (C |` y)) <-> <.y, (G` (C |` y))>. e. C))
50	44, 49	mpbird 196	. . . 4 |- ((dom F e. On /\ y = dom F) -> (C` y) = (G` (C |` y)))
51	50	ex 373	. . 3 |- (dom F e. On -> (y = dom F -> (C` y) = (G` (C |` y))))
52	25, 51	jaod 424	. 2 |- (dom F e. On -> ((y e. dom F \/ y = dom F) -> (C` y) = (G` (C |` y))))
53		elsuci 3025	. 2 |- (y e. suc dom F -> (y e. dom F \/ y = dom F))
54	52, 53	syl5 21	1 |- (dom F e. On -> (y e. suc dom F -> (C` y) = (G` (C |` y))))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456  A.wral 1637  E.wrex 1638   u. cun 2035   (_ wss 2037  {csn 2399  <.cop 2401  U.cuni 2493  Oncon0 2938  suc csuc 2940  dom cdm 3160   |` cres 3162  Fun wfun 3166   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  tfrlem12 3907

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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