Theorem numthlem 5945

Description: Lemma for numth 5946.

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 3487	. . 3 |- ~PA e. _V
3	2	ac4c 5913	. 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 5133	. . . . . . . . . 10 |- (x e. On -> (F` x) = (G` (F |` x)))
7	4, 5	tfrlem7 5125	. . . . . . . . . . . . 13 |- Fun F
8		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
9		resfunexg 4500	. . . . . . . . . . . . 13 |- ((Fun F /\ x e. _V) -> (F |` x) e. _V)
10	7, 8, 9	mp2an 761	. . . . . . . . . . . 12 |- (F |` x) e. _V
11		fvex 4689	. . . . . . . . . . . 12 |- (g` (A \ ran ( F |` x))) e. _V
12		rneq 4186	. . . . . . . . . . . . 13 |- (f = (F |` x) -> ran f = ran ( F |` x))
13		difeq2 2719	. . . . . . . . . . . . 13 |- (ran f = ran ( F |` x) -> (A \ ran f) = (A \ ran ( F |` x)))
14		fveq2 4681	. . . . . . . . . . . . 13 |- ((A \ ran f) = (A \ ran ( F |` x)) -> (g` (A \ ran f)) = (g` (A \ ran ( F |` x))))
15	12, 13, 14	3syl 24	. . . . . . . . . . . 12 |- (f = (F |` x) -> (g` (A \ ran f)) = (g` (A \ ran ( F |` x))))
16	10, 11, 15	fvopab 4753	. . . . . . . . . . 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 4682	. . . . . . . . . . 11 |- (G` (F |` x)) = ({<.f, y>. | y = (g` (A \ ran f))}` (F |` x))
19		df-ima 4007	. . . . . . . . . . . . 13 |- (F"x) = ran ( F |` x)
20	19	difeq2i 2723	. . . . . . . . . . . 12 |- (A \ (F"x)) = (A \ ran ( F |` x))
21	20	fveq2i 4684	. . . . . . . . . . 11 |- (g` (A \ (F"x))) = (g` (A \ ran ( F |` x)))
22	16, 18, 21	3eqtr4i 1921	. . . . . . . . . 10 |- (G` (F |` x)) = (g` (A \ (F"x)))
23	6, 22	syl6eq 1944	. . . . . . . . 9 |- (x e. On -> (F` x) = (g` (A \ (F"x))))
24	23	eleq1d 1963	. . . . . . . 8 |- (x e. On -> ((F` x) e. (A \ (F"x)) <-> (g` (A \ (F"x))) e. (A \ (F"x))))
25		difss 2735	. . . . . . . . . . 11 |- (A \ (F"x)) C_ A
26	1, 25	ssexi 3456	. . . . . . . . . . . 12 |- (A \ (F"x)) e. _V
27	26	elpw 3037	. . . . . . . . . . 11 |- ((A \ (F"x)) e. ~PA <-> (A \ (F"x)) C_ A)
28	25, 27	mpbir 207	. . . . . . . . . 10 |- (A \ (F"x)) e. ~PA
29		neeq1 2024	. . . . . . . . . . . 12 |- (y = (A \ (F"x)) -> (y =/= (/) <-> (A \ (F"x)) =/= (/)))
30		fveq2 4681	. . . . . . . . . . . . 13 |- (y = (A \ (F"x)) -> (g` y) = (g` (A \ (F"x))))
31		id 73	. . . . . . . . . . . . 13 |- (y = (A \ (F"x)) -> y = (A \ (F"x)))
32	30, 31	eleq12d 1965	. . . . . . . . . . . 12 |- (y = (A \ (F"x)) -> ((g` y) e. y <-> (g` (A \ (F"x))) e. (A \ (F"x))))
33	29, 32	imbi12d 688	. . . . . . . . . . 11 |- (y = (A \ (F"x)) -> ((y =/= (/) -> (g` y) e. y) <-> ((A \ (F"x)) =/= (/) -> (g` (A \ (F"x))) e. (A \ (F"x)))))
34	33	rcla4v 2376	. . . . . . . . . 10 |- ((A \ (F"x)) e. ~PA -> (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 377	. . . . . . . 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 227	. . . . . . 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 405	. . . . . 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 14	. . . . 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 2175	. . . 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 5132	. . . . 5 |- F Fn On
42	41, 1	tz7.49c 5169	. . . 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 4630	. . . . . 6 |- (f = (F |` x) -> (f:x-1-1-onto->A <-> (F |` x):x-1-1-onto->A))
44	10, 43	cla4ev 2371	. . . . 5 |- ((F |` x):x-1-1-onto->A -> E.f f:x-1-1-onto->A)
45	44	reximi 2198	. . . 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 24	. . 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 1674	. 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 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871   =/= wne 2017  A.wral 2105  E.wrex 2106  _Vcvv 2292   \ cdif 2590   C_ wss 2593  (/)c0 2875  ~Pcpw 3032  U.cuni 3177  {copab 3395  Oncon0 3657  ran crn 3987   |` cres 3988  "cima 3989  Fun wfun 3992   Fn wfn 3993  -1-1-onto->wf1o 3997  ` cfv 3998

This theorem is referenced by:  numth 5946

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-suc 3663  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014

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