Theorem inf3lemd 8524

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
inf3lemd	|- ( A e. om -> ( F ` A ) C_ x )

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 inf3lemd

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

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . 5 |- ( A = (/) -> ( F ` A ) = ( F ` (/) ) )
2		inf3lem.1	. . . . . 6 |- G = ( y e. _V |-> { w e. x | ( w i^i x ) C_ y } )
3		inf3lem.2	. . . . . 6 |- F = ( rec ( G , (/) ) |` om )
4		inf3lem.3	. . . . . 6 |- A e. _V
5		inf3lem.4	. . . . . 6 |- B e. _V
6	2, 3, 4, 5	inf3lemb 8522	. . . . 5 |- ( F ` (/) ) = (/)
7	1, 6	syl6eq 2672	. . . 4 |- ( A = (/) -> ( F ` A ) = (/) )
8		0ss 3972	. . . 4 |- (/) C_ x
9	7, 8	syl6eqss 3655	. . 3 |- ( A = (/) -> ( F ` A ) C_ x )
10	9	a1d 25	. 2 |- ( A = (/) -> ( A e. om -> ( F ` A ) C_ x ) )
11		nnsuc 7082	. . . 4 |- ( ( A e. om /\ A =/= (/) ) -> E. v e. om A = suc v )
12		vex 3203	. . . . . . . . . 10 |- v e. _V
13	2, 3, 12, 5	inf3lemc 8523	. . . . . . . . 9 |- ( v e. om -> ( F ` suc v ) = ( G ` ( F ` v ) ) )
14	13	eleq2d 2687	. . . . . . . 8 |- ( v e. om -> ( u e. ( F ` suc v ) <-> u e. ( G ` ( F ` v ) ) ) )
15		vex 3203	. . . . . . . . . 10 |- u e. _V
16		fvex 6201	. . . . . . . . . 10 |- ( F ` v ) e. _V
17	2, 3, 15, 16	inf3lema 8521	. . . . . . . . 9 |- ( u e. ( G ` ( F ` v ) ) <-> ( u e. x /\ ( u i^i x ) C_ ( F ` v ) ) )
18	17	simplbi 476	. . . . . . . 8 |- ( u e. ( G ` ( F ` v ) ) -> u e. x )
19	14, 18	syl6bi 243	. . . . . . 7 |- ( v e. om -> ( u e. ( F ` suc v ) -> u e. x ) )
20	19	ssrdv 3609	. . . . . 6 |- ( v e. om -> ( F ` suc v ) C_ x )
21		fveq2 6191	. . . . . . 7 |- ( A = suc v -> ( F ` A ) = ( F ` suc v ) )
22	21	sseq1d 3632	. . . . . 6 |- ( A = suc v -> ( ( F ` A ) C_ x <-> ( F ` suc v ) C_ x ) )
23	20, 22	syl5ibrcom 237	. . . . 5 |- ( v e. om -> ( A = suc v -> ( F ` A ) C_ x ) )
24	23	rexlimiv 3027	. . . 4 |- ( E. v e. om A = suc v -> ( F ` A ) C_ x )
25	11, 24	syl 17	. . 3 |- ( ( A e. om /\ A =/= (/) ) -> ( F ` A ) C_ x )
26	25	expcom 451	. 2 |- ( A =/= (/) -> ( A e. om -> ( F ` A ) C_ x ) )
27	10, 26	pm2.61ine 2877	1 |- ( A e. om -> ( F ` A ) C_ x )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   |-> cmpt 4729   |` cres 5116   suc csuc 5725   ` 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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506

This theorem is referenced by:  inf3lem2  8526  inf3lem3  8527  inf3lem6  8530