Theorem inf3lema 3460

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 3471 for detailed description.

Hypotheses

Ref	Expression
inf3lem.1	|- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
inf3lem.2	|- F = (rec(G, (/)) |` om)
inf3lem.3	|- A e. V
inf3lem.4	|- B e. V

Assertion

Ref	Expression
inf3lema	|- (A e. (G` B) <-> (A e. x /\ (A i^i x) (_ B))

Distinct variable group(s):   x,y,z,w

Proof of Theorem inf3lema

Step	Hyp	Ref	Expression
1		inf3lem.1	. . . . . 6 |- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
2		id 9	. . . . . . . 8 |- (z = u -> z = u)
3		sseq2 1522	. . . . . . . . . 10 |- (y = v -> ((w i^i x) (_ y <-> (w i^i x) (_ v))
4	3	birabsdv 1344	. . . . . . . . 9 |- (y = v -> {w e. x | (w i^i x) (_ y} = {w e. x | (w i^i x) (_ v})
5		ineq1 1638	. . . . . . . . . . 11 |- (w = f -> (w i^i x) = (f i^i x))
6	5	sseq1d 1527	. . . . . . . . . 10 |- (w = f -> ((w i^i x) (_ v <-> (f i^i x) (_ v))
7	6	cbvrabv 1426	. . . . . . . . 9 |- {w e. x | (w i^i x) (_ v} = {f e. x | (f i^i x) (_ v}
8	4, 7	syl6eq 1140	. . . . . . . 8 |- (y = v -> {w e. x | (w i^i x) (_ y} = {f e. x | (f i^i x) (_ v})
9	2, 8	cleqan12rd 1117	. . . . . . 7 |- ((y = v /\ z = u) -> (z = {w e. x | (w i^i x) (_ y} <-> u = {f e. x | (f i^i x) (_ v}))
10	9	cbvopabv 2105	. . . . . 6 |- {<.y, z>. | z = {w e. x | (w i^i x) (_ y}} = {<.v, u>. | u = {f e. x | (f i^i x) (_ v}}
11	1, 10	eqtr 1119	. . . . 5 |- G = {<.v, u>. | u = {f e. x | (f i^i x) (_ v}}
12	11	fveq1i 2833	. . . 4 |- (G` B) = ({<.v, u>. | u = {f e. x | (f i^i x) (_ v}}` B)
13		inf3lem.4	. . . . 5 |- B e. V
14		visset 1350	. . . . . 6 |- x e. V
15	14	rabex 1706	. . . . 5 |- {f e. x | (f i^i x) (_ B} e. V
16		sseq2 1522	. . . . . 6 |- (v = B -> ((f i^i x) (_ v <-> (f i^i x) (_ B))
17	16	birabsdv 1344	. . . . 5 |- (v = B -> {f e. x | (f i^i x) (_ v} = {f e. x | (f i^i x) (_ B})
18	13, 15, 17	fvopab 2877	. . . 4 |- ({<.v, u>. | u = {f e. x | (f i^i x) (_ v}}` B) = {f e. x | (f i^i x) (_ B}
19	12, 18	eqtr 1119	. . 3 |- (G` B) = {f e. x | (f i^i x) (_ B}
20	19	eleq2i 1153	. 2 |- (A e. (G` B) <-> A e. {f e. x | (f i^i x) (_ B})
21		ineq1 1638	. . . 4 |- (f = A -> (f i^i x) = (A i^i x))
22	21	sseq1d 1527	. . 3 |- (f = A -> ((f i^i x) (_ B <-> (A i^i x) (_ B))
23	22	elrab 1422	. 2 |- (A e. {f e. x | (f i^i x) (_ B} <-> (A e. x /\ (A i^i x) (_ B))
24	20, 23	bitr 151	1 |- (A e. (G` B) <-> (A e. x /\ (A i^i x) (_ B))

Syntax hints:   <-> wb 127   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  {crab 1204  Vcvv 1348   i^i cin 1486   (_ wss 1487  (/)c0 1707  {copab 2055  omcom 2372   |` cres 2412  ` cfv 2422  reccrdg 2969

This theorem is referenced by:  inf3lemd 3463  inf3lem1 3464  inf3lem2 3465  inf3lem3 3466

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-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-rab 1208  df-v 1349  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-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  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-fv 2438

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