Theorem frecabex 6563

Description: The class abstraction from df-frec 6556 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 4691	. . . 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 3302	. . . . . 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 2805	. . . . . . . 8 |- m e. _V
6		fvexg 5658	. . . . . . . 8 |- ( ( S e. V /\ m e. _V ) -> ( S ` m ) e. _V )
7	4, 5, 6	sylancl 413	. . . . . . 7 |- ( ph -> ( S ` m ) e. _V )
8		frecabex.fvex	. . . . . . 7 |- ( ph -> A. y ( F ` y ) e. _V )
9		fveq2 5639	. . . . . . . . 9 |- ( y = ( S ` m ) -> ( F ` y ) = ( F ` ( S ` m ) ) )
10	9	eleq1d 2300	. . . . . . . 8 |- ( y = ( S ` m ) -> ( ( F ` y ) e. _V <-> ( F ` ( S ` m ) ) e. _V ) )
11	10	spcgv 2893	. . . . . . 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 4228	. . . . . 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 414	. . . . 5 |- ( ph -> { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
15	14	ralrimivw 2606	. . . 4 |- ( ph -> A. m e. om { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
16		abrexex2g 6281	. . . 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 414	. . 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 3302	. . . 4 |- { x | ( dom S = (/) /\ x e. A ) } C_ A
20		frecabex.aex	. . . 4 |- ( ph -> A e. W )
21		ssexg 4228	. . . 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 414	. . 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 4539	. . 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 3474	. . . 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 2297	. . 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 )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 715   A.wal 1395   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   _Vcvv 2802   u. cun 3198   C_ wss 3200   (/)c0 3494   suc csuc 4462   omcom 4688   dom cdm 4725   ` cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334

This theorem is referenced by:  frectfr  6565