Theorem numthlem 3598

Description: Lemma for numth 3599.

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 group(s):   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 1806	. . 3 |- P~A e. V
3	2	ac4c 3572	. 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 2963	. . . . . . . . . 10 |- (x e. On -> (F` x) = (G` (F |` x)))
7		visset 1350	. . . . . . . . . . . . 13 |- x e. V
8	4, 5	tfrlem7 2955	. . . . . . . . . . . . 13 |- Fun F
9		resfunexg 2717	. . . . . . . . . . . . 13 |- (x e. V -> (Fun F -> (F |` x) e. V))
10	7, 8, 9	mp2 43	. . . . . . . . . . . 12 |- (F |` x) e. V
11		fvex 2838	. . . . . . . . . . . 12 |- (g` (A \ ran (F |` x))) e. V
12		rneq 2555	. . . . . . . . . . . . 13 |- (f = (F |` x) -> ran f = ran (F |` x))
13		difeq2 1583	. . . . . . . . . . . . 13 |- (ran f = ran (F |` x) -> (A \ ran f) = (A \ ran (F |` x)))
14		fveq2 2832	. . . . . . . . . . . . 13 |- ((A \ ran f) = (A \ ran (F |` x)) -> (g` (A \ ran f)) = (g` (A \ ran (F |` x))))
15	12, 13, 14	3syl 21	. . . . . . . . . . . 12 |- (f = (F |` x) -> (g` (A \ ran f)) = (g` (A \ ran (F |` x))))
16	10, 11, 15	fvopab 2877	. . . . . . . . . . 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 2833	. . . . . . . . . . 11 |- (G` (F |` x)) = ({<.f, y>. | y = (g` (A \ ran f))}` (F |` x))
19		df-ima 2431	. . . . . . . . . . . . 13 |- (F"x) = ran (F |` x)
20	19	difeq2i 1585	. . . . . . . . . . . 12 |- (A \ (F"x)) = (A \ ran (F |` x))
21	20	fveq2i 2835	. . . . . . . . . . 11 |- (g` (A \ (F"x))) = (g` (A \ ran (F |` x)))
22	16, 18, 21	3eqtr4 1126	. . . . . . . . . 10 |- (G` (F |` x)) = (g` (A \ (F"x)))
23	6, 22	syl6eq 1140	. . . . . . . . 9 |- (x e. On -> (F` x) = (g` (A \ (F"x))))
24	23	eleq1d 1155	. . . . . . . 8 |- (x e. On -> ((F` x) e. (A \ (F"x)) <-> (g` (A \ (F"x))) e. (A \ (F"x))))
25		difss 1596	. . . . . . . . . . 11 |- (A \ (F"x)) (_ A
26	1, 25	ssexi 1701	. . . . . . . . . . . 12 |- (A \ (F"x)) e. V
27	26	elpw 1801	. . . . . . . . . . 11 |- ((A \ (F"x)) e. P~A <-> (A \ (F"x)) (_ A)
28	25, 27	mpbir 165	. . . . . . . . . 10 |- (A \ (F"x)) e. P~A
29		cleq1 1107	. . . . . . . . . . . . 13 |- (y = (A \ (F"x)) -> (y = (/) <-> (A \ (F"x)) = (/)))
30	29	negbid 463	. . . . . . . . . . . 12 |- (y = (A \ (F"x)) -> (-. y = (/) <-> -. (A \ (F"x)) = (/)))
31		fveq2 2832	. . . . . . . . . . . . 13 |- (y = (A \ (F"x)) -> (g` y) = (g` (A \ (F"x))))
32		id 9	. . . . . . . . . . . . 13 |- (y = (A \ (F"x)) -> y = (A \ (F"x)))
33	31, 32	eleq12d 1157	. . . . . . . . . . . 12 |- (y = (A \ (F"x)) -> ((g` y) e. y <-> (g` (A \ (F"x))) e. (A \ (F"x))))
34	30, 33	imbi12d 474	. . . . . . . . . . 11 |- (y = (A \ (F"x)) -> ((-. y = (/) -> (g` y) e. y) <-> (-. (A \ (F"x)) = (/) -> (g` (A \ (F"x))) e. (A \ (F"x)))))
35	34	rcla4v 1402	. . . . . . . . . 10 |- (A.y e. P~ A(-. y = (/) -> (g` y) e. y) -> ((A \ (F"x)) e. P~A -> (-. (A \ (F"x)) = (/) -> (g` (A \ (F"x))) e. (A \ (F"x)))))
36	28, 35	mpi 44	. . . . . . . . 9 |- (A.y e. P~ A(-. y = (/) -> (g` y) e. y) -> (-. (A \ (F"x)) = (/) -> (g` (A \ (F"x))) e. (A \ (F"x))))
37	36	imp 277	. . . . . . . 8 |- ((A.y e. P~ A(-. y = (/) -> (g` y) e. y) /\ -. (A \ (F"x)) = (/)) -> (g` (A \ (F"x))) e. (A \ (F"x)))
38	24, 37	syl5bir 184	. . . . . . 7 |- (x e. On -> ((A.y e. P~ A(-. y = (/) -> (g` y) e. y) /\ -. (A \ (F"x)) = (/)) -> (F` x) e. (A \ (F"x))))
39	38	exp3a 292	. . . . . 6 |- (x e. On -> (A.y e. P~ A(-. y = (/) -> (g` y) e. y) -> (-. (A \ (F"x)) = (/) -> (F` x) e. (A \ (F"x)))))
40	39	com12 13	. . . . 5 |- (A.y e. P~ A(-. y = (/) -> (g` y) e. y) -> (x e. On -> (-. (A \ (F"x)) = (/) -> (F` x) e. (A \ (F"x)))))
41	40	r19.21aiv 1259	. . . 4 |- (A.y e. P~ A(-. y = (/) -> (g` y) e. y) -> A.x e. On (-. (A \ (F"x)) = (/) -> (F` x) e. (A \ (F"x))))
42	4, 5	tfr1 2962	. . . . 5 |- F Fn On
43	42, 1	tz7.49c 2998	. . . 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)
44		f1oeq1 2795	. . . . . 6 |- (f = (F |` x) -> (f:x-1-1-onto->A <-> (F |` x):x-1-1-onto->A))
45	10, 44	cla4ev 1401	. . . . 5 |- ((F |` x):x-1-1-onto->A -> E.f f:x-1-1-onto->A)
46	45	r19.22si 1275	. . . 4 |- (E.x e. On (F |` x):x-1-1-onto->A -> E.x e. On E.f f:x-1-1-onto->A)
47	41, 43, 46	3syl 21	. . 3 |- (A.y e. P~ A(-. y = (/) -> (g` y) e. y) -> E.x e. On E.f f:x-1-1-onto->A)
48	47	19.23aiv 952	. 2 |- (E.gA.y e. P~ A(-. y = (/) -> (g` y) e. y) -> E.x e. On E.f f:x-1-1-onto->A)
49	3, 48	ax-mp 6	1 |- E.x e. On E.f f:x-1-1-onto->A

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  Vcvv 1348   \ cdif 1484   (_ wss 1487  (/)c0 1707  P~cpw 1798  U.cuni 1919  {copab 2055  Oncon0 2199  ran crn 2411   |` cres 2412  "cima 2413  Fun wfun 2416   Fn wfn 2417  -1-1-onto->wf1o 2421  ` cfv 2422

This theorem is referenced by:  numth 3599

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438

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