Theorem inf3lema 4609

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 4620 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:   x,y,z,w

Proof of Theorem inf3lema

Step	Hyp	Ref	Expression
1		ineq1 2210	. . 3 |- (f = A -> (f i^i x) = (A i^i x))
2	1	sseq1d 2088	. 2 |- (f = A -> ((f i^i x) (_ B <-> (A i^i x) (_ B))
3		inf3lem.1	. . . . 5 |- G = {<.y, z>. | z = {w e. x | (w i^i x) (_ y}}
4		id 59	. . . . . . 7 |- (z = u -> z = u)
5		sseq2 2083	. . . . . . . . 9 |- (y = v -> ((w i^i x) (_ y <-> (w i^i x) (_ v))
6	5	rabbisdv 1807	. . . . . . . 8 |- (y = v -> {w e. x | (w i^i x) (_ y} = {w e. x | (w i^i x) (_ v})
7		ineq1 2210	. . . . . . . . . 10 |- (w = f -> (w i^i x) = (f i^i x))
8	7	sseq1d 2088	. . . . . . . . 9 |- (w = f -> ((w i^i x) (_ v <-> (f i^i x) (_ v))
9	8	cbvrabv 1911	. . . . . . . 8 |- {w e. x | (w i^i x) (_ v} = {f e. x | (f i^i x) (_ v}
10	6, 9	syl6eq 1523	. . . . . . 7 |- (y = v -> {w e. x | (w i^i x) (_ y} = {f e. x | (f i^i x) (_ v})
11	4, 10	eqeqan12rd 1491	. . . . . 6 |- ((y = v /\ z = u) -> (z = {w e. x | (w i^i x) (_ y} <-> u = {f e. x | (f i^i x) (_ v}))
12	11	cbvopabv 2673	. . . . 5 |- {<.y, z>. | z = {w e. x | (w i^i x) (_ y}} = {<.v, u>. | u = {f e. x | (f i^i x) (_ v}}
13	3, 12	eqtr 1495	. . . 4 |- G = {<.v, u>. | u = {f e. x | (f i^i x) (_ v}}
14	13	fveq1i 3725	. . 3 |- (G` B) = ({<.v, u>. | u = {f e. x | (f i^i x) (_ v}}` B)
15		inf3lem.4	. . . 4 |- B e. V
16		visset 1813	. . . . 5 |- x e. V
17	16	rabex 2725	. . . 4 |- {f e. x | (f i^i x) (_ B} e. V
18		sseq2 2083	. . . . 5 |- (v = B -> ((f i^i x) (_ v <-> (f i^i x) (_ B))
19	18	rabbisdv 1807	. . . 4 |- (v = B -> {f e. x | (f i^i x) (_ v} = {f e. x | (f i^i x) (_ B})
20	15, 17, 19	fvopab 3790	. . 3 |- ({<.v, u>. | u = {f e. x | (f i^i x) (_ v}}` B) = {f e. x | (f i^i x) (_ B}
21	14, 20	eqtr 1495	. 2 |- (G` B) = {f e. x | (f i^i x) (_ B}
22	2, 21	elrab2 1907	1 |- (A e. (G` B) <-> (A e. x /\ (A i^i x) (_ B))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  {crab 1648  Vcvv 1811   i^i cin 2046   (_ wss 2047  (/)c0 2280  {copab 2666  omcom 3131   |` cres 3172  ` cfv 3182  reccrdg 3931

This theorem is referenced by:  inf3lemd 4612  inf3lem1 4613  inf3lem2 4614  inf3lem3 4615

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198

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