Theorem frecabex 6669

Description: The class abstraction from df-frec 6662 exists. This is a lemma for other finite recursion proofs. (Contributed by Jim Kingdon, 13-May-2020.)

Hypotheses

Ref	Expression
frecabex.sex	|- ( ph -> S e. V )
frecabex.fvex	|- ( ph -> A. y ( F ` y ) e. _V )
frecabex.aex	|- ( ph -> A e. W )

Assertion

Ref	Expression
frecabex	|- ( ph -> { x | ( E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) \/ ( dom S = (/) /\ x e. A ) ) } e. _V )

Distinct variable groups:   x, A   x, F   x, S, y   ph, m   x, m, y   y, F

Allowed substitution hints:   ph( x, y)   A( y, m)   S( m)   F( m)   V( x, y, m)   W( x, y, m)

Proof of Theorem frecabex

Step	Hyp	Ref	Expression
1		omex 4740	. . . 4 |- om e. _V
2		simpr 110	. . . . . . 7 |- ( ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) -> x e. ( F ` ( S ` m ) ) )
3	2	abssi 3323	. . . . . 6 |- { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } C_ ( F ` ( S ` m ) )
4		frecabex.sex	. . . . . . . 8 |- ( ph -> S e. V )
5		vex 2824	. . . . . . . 8 |- m e. _V
6		fvexg 5714	. . . . . . . 8 |- ( ( S e. V /\ m e. _V ) -> ( S ` m ) e. _V )
7	4, 5, 6	sylancl 417	. . . . . . 7 |- ( ph -> ( S ` m ) e. _V )
8		frecabex.fvex	. . . . . . 7 |- ( ph -> A. y ( F ` y ) e. _V )
9		fveq2 5695	. . . . . . . . 9 |- ( y = ( S ` m ) -> ( F ` y ) = ( F ` ( S ` m ) ) )
10	9	eleq1d 2307	. . . . . . . 8 |- ( y = ( S ` m ) -> ( ( F ` y ) e. _V <-> ( F ` ( S ` m ) ) e. _V ) )
11	10	spcgv 2912	. . . . . . 7 |- ( ( S ` m ) e. _V -> ( A. y ( F ` y ) e. _V -> ( F ` ( S ` m ) ) e. _V ) )
12	7, 8, 11	sylc 62	. . . . . 6 |- ( ph -> ( F ` ( S ` m ) ) e. _V )
13		ssexg 4272	. . . . . 6 |- ( ( { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } C_ ( F ` ( S ` m ) ) /\ ( F ` ( S ` m ) ) e. _V ) -> { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
14	3, 12, 13	sylancr 418	. . . . 5 |- ( ph -> { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
15	14	ralrimivw 2624	. . . 4 |- ( ph -> A. m e. om { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
16		abrexex2g 6349	. . . 4 |- ( ( om e. _V /\ A. m e. om { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V ) -> { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
17	1, 15, 16	sylancr 418	. . 3 |- ( ph -> { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
18		simpr 110	. . . . 5 |- ( ( dom S = (/) /\ x e. A ) -> x e. A )
19	18	abssi 3323	. . . 4 |- { x | ( dom S = (/) /\ x e. A ) } C_ A
20		frecabex.aex	. . . 4 |- ( ph -> A e. W )
21		ssexg 4272	. . . 4 |- ( ( { x | ( dom S = (/) /\ x e. A ) } C_ A /\ A e. W ) -> { x | ( dom S = (/) /\ x e. A ) } e. _V )
22	19, 20, 21	sylancr 418	. . 3 |- ( ph -> { x | ( dom S = (/) /\ x e. A ) } e. _V )
23	17, 22	jca 306	. 2 |- ( ph -> ( { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V /\ { x | ( dom S = (/) /\ x e. A ) } e. _V ) )
24		unexb 4588	. . 3 |- ( ( { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V /\ { x | ( dom S = (/) /\ x e. A ) } e. _V ) <-> ( { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } u. { x | ( dom S = (/) /\ x e. A ) } ) e. _V )
25		unab 3498	. . . 4 |- ( { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } u. { x | ( dom S = (/) /\ x e. A ) } ) = { x | ( E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) \/ ( dom S = (/) /\ x e. A ) ) }
26	25	eleq1i 2304	. . 3 |- ( ( { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } u. { x | ( dom S = (/) /\ x e. A ) } ) e. _V <-> { x | ( E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) \/ ( dom S = (/) /\ x e. A ) ) } e. _V )
27	24, 26	bitri 184	. 2 |- ( ( { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V /\ { x | ( dom S = (/) /\ x e. A ) } e. _V ) <-> { x | ( E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) \/ ( dom S = (/) /\ x e. A ) ) } e. _V )
28	23, 27	sylib 122	1 |- ( ph -> { x | ( E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) \/ ( dom S = (/) /\ x e. A ) ) } e. _V )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   \/ wo 720   A.wal 1400   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   _Vcvv 2821   u. cun 3218   C_ wss 3220   (/)c0 3520   suc csuc 4510   omcom 4737   dom cdm 4774   ` cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385

This theorem is used by:  frectfr  6671