Theorem numthlem 4766

Description: Lemma for numth 4767.

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 2741	. . 3 |- P~A e. V
3	2	ac4c 4734	. 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 3920	. . . . . . . . . 10 |- (x e. On -> (F` x) = (G` (F |` x)))
7	4, 5	tfrlem7 3912	. . . . . . . . . . . . 13 |- Fun F
8		visset 1810	. . . . . . . . . . . . 13 |- x e. V
9		resfunexg 3575	. . . . . . . . . . . . 13 |- ((Fun F /\ x e. V) -> (F |` x) e. V)
10	7, 8, 9	mp2an 696	. . . . . . . . . . . 12 |- (F |` x) e. V
11		fvex 3727	. . . . . . . . . . . 12 |- (g` (A \ ran ( F |` x))) e. V
12		rneq 3335	. . . . . . . . . . . . 13 |- (f = (F |` x) -> ran f = ran ( F |` x))
13		difeq2 2151	. . . . . . . . . . . . 13 |- (ran f = ran ( F |` x) -> (A \ ran f) = (A \ ran ( F |` x)))
14		fveq2 3719	. . . . . . . . . . . . 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 3785	. . . . . . . . . . 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 3720	. . . . . . . . . . 11 |- (G` (F |` x)) = ({<.f, y>. | y = (g` (A \ ran f))}` (F |` x))
19		df-ima 3187	. . . . . . . . . . . . 13 |- (F"x) = ran ( F |` x)
20	19	difeq2i 2153	. . . . . . . . . . . 12 |- (A \ (F"x)) = (A \ ran ( F |` x))
21	20	fveq2i 3722	. . . . . . . . . . 11 |- (g` (A \ (F"x))) = (g` (A \ ran ( F |` x)))
22	16, 18, 21	3eqtr4 1503	. . . . . . . . . 10 |- (G` (F |` x)) = (g` (A \ (F"x)))
23	6, 22	syl6eq 1521	. . . . . . . . 9 |- (x e. On -> (F` x) = (g` (A \ (F"x))))
24	23	eleq1d 1538	. . . . . . . 8 |- (x e. On -> ((F` x) e. (A \ (F"x)) <-> (g` (A \ (F"x))) e. (A \ (F"x))))
25		difss 2164	. . . . . . . . . . 11 |- (A \ (F"x)) (_ A
26	1, 25	ssexi 2716	. . . . . . . . . . . 12 |- (A \ (F"x)) e. V
27	26	elpw 2401	. . . . . . . . . . 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 1588	. . . . . . . . . . . 12 |- (y = (A \ (F"x)) -> (y =/= (/) <-> (A \ (F"x)) =/= (/)))
30		fveq2 3719	. . . . . . . . . . . . 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 1540	. . . . . . . . . . . 12 |- (y = (A \ (F"x)) -> ((g` y) e. y <-> (g` (A \ (F"x))) e. (A \ (F"x))))
33	29, 32	imbi12d 625	. . . . . . . . . . 11 |- (y = (A \ (F"x)) -> ((y =/= (/) -> (g` y) e. y) <-> ((A \ (F"x)) =/= (/) -> (g` (A \ (F"x))) e. (A \ (F"x)))))
34	33	rcla4v 1870	. . . . . . . . . 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 1711	. . . 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 3919	. . . . 5 |- F Fn On
42	41, 1	tz7.49c 3955	. . . 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 3679	. . . . . 6 |- (f = (F |` x) -> (f:x-1-1-onto->A <-> (F |` x):x-1-1-onto->A))
44	10, 43	cla4ev 1866	. . . . 5 |- ((F |` x):x-1-1-onto->A -> E.f f:x-1-1-onto->A)
45	44	r19.22si 1732	. . . 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 1294	. 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 955   e. wcel 957  E.wex 979  {cab 1462   =/= wne 1583  A.wral 1643  E.wrex 1644  Vcvv 1808   \ cdif 2041   (_ wss 2044  (/)c0 2277  P~cpw 2398  U.cuni 2499  {copab 2662  Oncon0 2944  ran crn 3167   |` cres 3168  "cima 3169  Fun wfun 3172   Fn wfn 3173  -1-1-onto->wf1o 3177  ` cfv 3178

This theorem is referenced by:  numth 4767

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-tp 2412  df-op 2413  df-uni 2500  df-int 2530  df-iun 2564  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-id 2831  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947  df-on 2948  df-suc 2950  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193  df-fv 3194

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