Theorem inf3lema 8521

Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8532 for detailed description. (Contributed by NM, 28-Oct-1996.)

Hypotheses

Ref	Expression
inf3lem.1	|- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ 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 ) C_ B ) )

Distinct variable group:   x, y, w

Allowed substitution hints:   A( x, y, w)   B( x, y, w)   F( x, y, w)   G( x, y, w)

Proof of Theorem inf3lema

Dummy variables v f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ineq1 3807	. . 3 |- ( f = A -> ( f i^i x ) = ( A i^i x ) )
2	1	sseq1d 3632	. 2 |- ( f = A -> ( ( f i^i x ) C_ B <-> ( A i^i x ) C_ B ) )
3		inf3lem.4	. . 3 |- B e. _V
4		sseq2 3627	. . . . 5 |- ( v = B -> ( ( f i^i x ) C_ v <-> ( f i^i x ) C_ B ) )
5	4	rabbidv 3189	. . . 4 |- ( v = B -> { f e. x | ( f i^i x ) C_ v } = { f e. x | ( f i^i x ) C_ B } )
6		inf3lem.1	. . . . 5 |- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
7		sseq2 3627	. . . . . . . 8 |- ( y = v -> ( ( w i^i x ) C_ y <-> ( w i^i x ) C_ v ) )
8	7	rabbidv 3189	. . . . . . 7 |- ( y = v -> { w e. x | ( w i^i x ) C_ y } = { w e. x | ( w i^i x ) C_ v } )
9		ineq1 3807	. . . . . . . . 9 |- ( w = f -> ( w i^i x ) = ( f i^i x ) )
10	9	sseq1d 3632	. . . . . . . 8 |- ( w = f -> ( ( w i^i x ) C_ v <-> ( f i^i x ) C_ v ) )
11	10	cbvrabv 3199	. . . . . . 7 |- { w e. x | ( w i^i x ) C_ v } = { f e. x | ( f i^i x ) C_ v }
12	8, 11	syl6eq 2672	. . . . . 6 |- ( y = v -> { w e. x | ( w i^i x ) C_ y } = { f e. x | ( f i^i x ) C_ v } )
13	12	cbvmptv 4750	. . . . 5 |- ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } ) = ( v e. _V |-> { f e. x | ( f i^i x ) C_ v } )
14	6, 13	eqtri 2644	. . . 4 |- G = ( v e. _V |-> { f e. x | ( f i^i x ) C_ v } )
15		vex 3203	. . . . 5 |- x e. _V
16	15	rabex 4813	. . . 4 |- { f e. x | ( f i^i x ) C_ B } e. _V
17	5, 14, 16	fvmpt 6282	. . 3 |- ( B e. _V -> ( G ` B ) = { f e. x | ( f i^i x ) C_ B } )
18	3, 17	ax-mp 5	. 2 |- ( G ` B ) = { f e. x | ( f i^i x ) C_ B }
19	2, 18	elrab2 3366	1 |- ( A e. ( G ` B ) <-> ( A e. x /\ ( A i^i x ) C_ B ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   |` cres 5116   ` cfv 5888   omcom 7065   reccrdg 7505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  inf3lemd  8524  inf3lem1  8525  inf3lem2  8526  inf3lem3  8527