Theorem tz7.48-2 3942

Description: Proposition 7.48(2) of [TakeutiZaring] p. 51.

Hypothesis

Ref	Expression
tz7.48.1	|- F Fn On

Assertion

Ref	Expression
tz7.48-2	|- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)

Distinct variable groups:   x,F   x,A

Proof of Theorem tz7.48-2

Step	Hyp	Ref	Expression
1		onelon 2962	. . . . . . . . . 10 |- ((x e. On /\ y e. x) -> y e. On)
2	1	ancoms 436	. . . . . . . . 9 |- ((y e. x /\ x e. On) -> y e. On)
3		dmres 3364	. . . . . . . . . . . . . . 15 |- dom ( F |` x) = (x i^i dom F)
4	3	eleq2i 1530	. . . . . . . . . . . . . 14 |- (y e. dom ( F |` x) <-> y e. (x i^i dom F))
5		elin 2197	. . . . . . . . . . . . . 14 |- (y e. (x i^i dom F) <-> (y e. x /\ y e. dom F))
6	4, 5	bitr 173	. . . . . . . . . . . . 13 |- (y e. dom ( F |` x) <-> (y e. x /\ y e. dom F))
7		tz7.48.1	. . . . . . . . . . . . . . . 16 |- F Fn On
8		fnfun 3571	. . . . . . . . . . . . . . . 16 |- (F Fn On -> Fun F)
9	7, 8	ax-mp 7	. . . . . . . . . . . . . . 15 |- Fun F
10		funres 3537	. . . . . . . . . . . . . . 15 |- (Fun F -> Fun (F |` x))
11	9, 10	ax-mp 7	. . . . . . . . . . . . . 14 |- Fun (F |` x)
12		fvelrn 3797	. . . . . . . . . . . . . 14 |- ((Fun (F |` x) /\ y e. dom ( F |` x)) -> ((F |` x)` y) e. ran ( F |` x))
13	11, 12	mpan 693	. . . . . . . . . . . . 13 |- (y e. dom ( F |` x) -> ((F |` x)` y) e. ran ( F |` x))
14	6, 13	sylbir 201	. . . . . . . . . . . 12 |- ((y e. x /\ y e. dom F) -> ((F |` x)` y) e. ran ( F |` x))
15		fvres 3719	. . . . . . . . . . . . . . 15 |- (y e. x -> ((F |` x)` y) = (F` y))
16	15	eleq1d 1532	. . . . . . . . . . . . . 14 |- (y e. x -> (((F |` x)` y) e. ran ( F |` x) <-> (F` y) e. ran ( F |` x)))
17		df-ima 3181	. . . . . . . . . . . . . . 15 |- (F"x) = ran ( F |` x)
18	17	eleq2i 1530	. . . . . . . . . . . . . 14 |- ((F` y) e. (F"x) <-> (F` y) e. ran ( F |` x))
19	16, 18	syl6rbbr 537	. . . . . . . . . . . . 13 |- (y e. x -> ((F` y) e. (F"x) <-> ((F |` x)` y) e. ran ( F |` x)))
20	19	adantr 389	. . . . . . . . . . . 12 |- ((y e. x /\ y e. dom F) -> ((F` y) e. (F"x) <-> ((F |` x)` y) e. ran ( F |` x)))
21	14, 20	mpbird 196	. . . . . . . . . . 11 |- ((y e. x /\ y e. dom F) -> (F` y) e. (F"x))
22		eleq1a 1535	. . . . . . . . . . . 12 |- ((F` y) e. (F"x) -> ((F` x) = (F` y) -> (F` x) e. (F"x)))
23		eldifn 2153	. . . . . . . . . . . 12 |- ((F` x) e. (A \ (F"x)) -> -. (F` x) e. (F"x))
24	22, 23	nsyli 121	. . . . . . . . . . 11 |- ((F` y) e. (F"x) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
25	21, 24	syl 10	. . . . . . . . . 10 |- ((y e. x /\ y e. dom F) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
26		fndm 3573	. . . . . . . . . . . 12 |- (F Fn On -> dom F = On)
27	7, 26	ax-mp 7	. . . . . . . . . . 11 |- dom F = On
28	27	eleq2i 1530	. . . . . . . . . 10 |- (y e. dom F <-> y e. On)
29	25, 28	sylan2br 453	. . . . . . . . 9 |- ((y e. x /\ y e. On) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
30	2, 29	syldan 467	. . . . . . . 8 |- ((y e. x /\ x e. On) -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y)))
31	30	ex 373	. . . . . . 7 |- (y e. x -> (x e. On -> ((F` x) e. (A \ (F"x)) -> -. (F` x) = (F` y))))
32	31	imp3a 361	. . . . . 6 |- (y e. x -> ((x e. On /\ (F` x) e. (A \ (F"x))) -> -. (F` x) = (F` y)))
33	32	com12 11	. . . . 5 |- ((x e. On /\ (F` x) e. (A \ (F"x))) -> (y e. x -> -. (F` x) = (F` y)))
34	33	r19.21aiv 1705	. . . 4 |- ((x e. On /\ (F` x) e. (A \ (F"x))) -> A.y e. x -. (F` x) = (F` y))
35	34	r19.20ia 1697	. . 3 |- (A.x e. On (F` x) e. (A \ (F"x)) -> A.x e. On A.y e. x -. (F` x) = (F` y))
36		ssid 2070	. . . 4 |- On (_ On
37	7	tz7.48lem 3940	. . . 4 |- ((On (_ On /\ A.x e. On A.y e. x -. (F` x) = (F` y)) -> Fun `'(F |` On))
38	36, 37	mpan 693	. . 3 |- (A.x e. On A.y e. x -. (F` x) = (F` y) -> Fun `'(F |` On))
39	35, 38	syl 10	. 2 |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'(F |` On))
40		fnrel 3572	. . . . . 6 |- (F Fn On -> Rel F)
41	7, 40	ax-mp 7	. . . . 5 |- Rel F
42	27	eqimssi 2101	. . . . 5 |- dom F (_ On
43		relssres 3376	. . . . 5 |- ((Rel F /\ dom F (_ On) -> (F |` On) = F)
44	41, 42, 43	mp2an 695	. . . 4 |- (F |` On) = F
45		cnveq 3281	. . . 4 |- ((F |` On) = F -> `'(F |` On) = `'F)
46	44, 45	ax-mp 7	. . 3 |- `'(F |` On) = `'F
47		funeq 3521	. . 3 |- (`'(F |` On) = `'F -> (Fun `'(F |` On) <-> Fun `'F))
48	46, 47	ax-mp 7	. 2 |- (Fun `'(F |` On) <-> Fun `'F)
49	39, 48	sylib 198	1 |- (A.x e. On (F` x) e. (A \ (F"x)) -> Fun `'F)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637   \ cdif 2034   i^i cin 2036   (_ wss 2037  Oncon0 2938  `'ccnv 3159  dom cdm 3160  ran crn 3161   |` cres 3162  "cima 3163  Rel wrel 3165  Fun wfun 3166   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  tz7.48-3 3943

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-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-v 1803  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-op 2406  df-uni 2494  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-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-f 3184  df-f1 3185  df-fv 3188

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