Theorem numthlem 4755

Description: Lemma for numth 4756.

Hypotheses

Ref	Expression
numthlem.1	|- A e. V
numthlem.2	|- B = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
numthlem.3	|- F = U.B
numthlem.4	|- G = {<.f, y>. | y = (g` (A \ ran f))}

Assertion

Ref	Expression
numthlem	|- E.x e. On E.f f:x-1-1-onto->A

Distinct variable groups:   x,y,f,g,A   x,B,y,f   x,F,y,f   x,G,y,f

Proof of Theorem numthlem

Step	Hyp	Ref	Expression
1		numthlem.1	. . . 4 |- A e. V
2	1	pwex 2735	. . 3 |- P~A e. V
3	2	ac4c 4723	. 2 |- E.gA.y e. P~ A(y =/= (/) -> (g` y) e. y)
4		numthlem.2	. . . . . . . . . . 11 |- B = {f | E.x e. On (f Fn x /\ A.y e. x (f` y) = (G` (f |` y)))}
5		numthlem.3	. . . . . . . . . . 11 |- F = U.B
6	4, 5	tfr2 3910	. . . . . . . . . 10 |- (x e. On -> (F` x) = (G` (F |` x)))
7	4, 5	tfrlem7 3902	. . . . . . . . . . . . 13 |- Fun F
8		visset 1804	. . . . . . . . . . . . 13 |- x e. V
9		resfunexg 3565	. . . . . . . . . . . . 13 |- ((Fun F /\ x e. V) -> (F |` x) e. V)
10	7, 8, 9	mp2an 695	. . . . . . . . . . . 12 |- (F |` x) e. V
11		fvex 3717	. . . . . . . . . . . 12 |- (g` (A \ ran ( F |` x))) e. V
12		rneq 3328	. . . . . . . . . . . . 13 |- (f = (F |` x) -> ran f = ran ( F |` x))
13		difeq2 2144	. . . . . . . . . . . . 13 |- (ran f = ran ( F |` x) -> (A \ ran f) = (A \ ran ( F |` x)))
14		fveq2 3709	. . . . . . . . . . . . 13 |- ((A \ ran f) = (A \ ran ( F |` x)) -> (g` (A \ ran f)) = (g` (A \ ran ( F |` x))))
15	12, 13, 14	3syl 20	. . . . . . . . . . . 12 |- (f = (F |` x) -> (g` (A \ ran f)) = (g` (A \ ran ( F |` x))))
16	10, 11, 15	fvopab 3775	. . . . . . . . . . 11 |- ({<.f, y>. | y = (g` (A \ ran f))}` (F |` x)) = (g` (A \ ran ( F |` x)))
17		numthlem.4	. . . . . . . . . . . 12 |- G = {<.f, y>. | y = (g` (A \ ran f))}
18	17	fveq1i 3710	. . . . . . . . . . 11 |- (G` (F |` x)) = ({<.f, y>. | y = (g` (A \ ran f))}` (F |` x))
19		df-ima 3181	. . . . . . . . . . . . 13 |- (F"x) = ran ( F |` x)
20	19	difeq2i 2146	. . . . . . . . . . . 12 |- (A \ (F"x)) = (A \ ran ( F |` x))
21	20	fveq2i 3712	. . . . . . . . . . 11 |- (g` (A \ (F"x))) = (g` (A \ ran ( F |` x)))
22	16, 18, 21	3eqtr4 1497	. . . . . . . . . 10 |- (G` (F |` x)) = (g` (A \ (F"x)))
23	6, 22	syl6eq 1515	. . . . . . . . 9 |- (x e. On -> (F` x) = (g` (A \ (F"x))))
24	23	eleq1d 1532	. . . . . . . 8 |- (x e. On -> ((F` x) e. (A \ (F"x)) <-> (g` (A \ (F"x))) e. (A \ (F"x))))
25		difss 2157	. . . . . . . . . . 11 |- (A \ (F"x)) (_ A
26	1, 25	ssexi 2710	. . . . . . . . . . . 12 |- (A \ (F"x)) e. V
27	26	elpw 2394	. . . . . . . . . . 11 |- ((A \ (F"x)) e. P~A <-> (A \ (F"x)) (_ A)
28	25, 27	mpbir 190	. . . . . . . . . 10 |- (A \ (F"x)) e. P~A
29		neeq1 1582	. . . . . . . . . . . 12 |- (y = (A \ (F"x)) -> (y =/= (/) <-> (A \ (F"x)) =/= (/)))
30		fveq2 3709	. . . . . . . . . . . . 13 |- (y = (A \ (F"x)) -> (g` y) = (g` (A \ (F"x))))
31		id 59	. . . . . . . . . . . . 13 |- (y = (A \ (F"x)) -> y = (A \ (F"x)))
32	30, 31	eleq12d 1534	. . . . . . . . . . . 12 |- (y = (A \ (F"x)) -> ((g` y) e. y <-> (g` (A \ (F"x))) e. (A \ (F"x))))
33	29, 32	imbi12d 624	. . . . . . . . . . 11 |- (y = (A \ (F"x)) -> ((y =/= (/) -> (g` y) e. y) <-> ((A \ (F"x)) =/= (/) -> (g` (A \ (F"x))) e. (A \ (F"x)))))
34	33	rcla4v 1864	. . . . . . . . . 10 |- ((A \ (F"x)) e. P~A -> (A.y e. P~ A(y =/= (/) -> (g` y) e. y) -> ((A \ (F"x)) =/= (/) -> (g` (A \ (F"x))) e. (A \ (F"x)))))
35	28, 34	ax-mp 7	. . . . . . . . 9 |- (A.y e. P~ A(y =/= (/) -> (g` y) e. y) -> ((A \ (F"x)) =/= (/) -> (g` (A \ (F"x))) e. (A \ (F"x))))
36	35	imp 350	. . . . . . . 8 |- ((A.y e. P~ A(y =/= (/) -> (g` y) e. y) /\ (A \ (F"x)) =/= (/)) -> (g` (A \ (F"x))) e. (A \ (F"x)))
37	24, 36	syl5bir 210	. . . . . . 7 |- (x e. On -> ((A.y e. P~ A(y =/= (/) -> (g` y) e. y) /\ (A \ (F"x)) =/= (/)) -> (F` x) e. (A \ (F"x))))
38	37	exp3a 375	. . . . . 6 |- (x e. On -> (A.y e. P~ A(y =/= (/) -> (g` y) e. y) -> ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x)))))
39	38	com12 11	. . . . 5 |- (A.y e. P~ A(y =/= (/) -> (g` y) e. y) -> (x e. On -> ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x)))))
40	39	r19.21aiv 1705	. . . 4 |- (A.y e. P~ A(y =/= (/) -> (g` y) e. y) -> A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))))
41	4, 5	tfr1 3909	. . . . 5 |- F Fn On
42	41, 1	tz7.49c 3945	. . . 4 |- (A.x e. On ((A \ (F"x)) =/= (/) -> (F` x) e. (A \ (F"x))) -> E.x e. On (F |` x):x-1-1-onto->A)
43		f1oeq1 3669	. . . . . 6 |- (f = (F |` x) -> (f:x-1-1-onto->A <-> (F |` x):x-1-1-onto->A))
44	10, 43	cla4ev 1860	. . . . 5 |- ((F |` x):x-1-1-onto->A -> E.f f:x-1-1-onto->A)
45	44	r19.22si 1726	. . . 4 |- (E.x e. On (F |` x):x-1-1-onto->A -> E.x e. On E.f f:x-1-1-onto->A)
46	40, 42, 45	3syl 20	. . 3 |- (A.y e. P~ A(y =/= (/) -> (g` y) e. y) -> E.x e. On E.f f:x-1-1-onto->A)
47	46	19.23aiv 1290	. 2 |- (E.gA.y e. P~ A(y =/= (/) -> (g` y) e. y) -> E.x e. On E.f f:x-1-1-onto->A)
48	3, 47	ax-mp 7	1 |- E.x e. On E.f f:x-1-1-onto->A

Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  {cab 1456   =/= wne 1577  A.wral 1637  E.wrex 1638  Vcvv 1802   \ cdif 2034   (_ wss 2037  (/)c0 2270  P~cpw 2391  U.cuni 2493  {copab 2656  Oncon0 2938  ran crn 3161   |` cres 3162  "cima 3163  Fun wfun 3166   Fn wfn 3167  -1-1-onto->wf1o 3171  ` cfv 3172

This theorem is referenced by:  numth 4756

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-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-ac 4716

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-reu 1643  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-int 2524  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-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188

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