Theorem frecabex 6366

Description: The class abstraction from df-frec 6359 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 4570	. . . 4 |- om e. _V
2		simpr 109	. . . . . . 7 |- ( ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) -> x e. ( F ` ( S ` m ) ) )
3	2	abssi 3217	. . . . . 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 2729	. . . . . . . 8 |- m e. _V
6		fvexg 5505	. . . . . . . 8 |- ( ( S e. V /\ m e. _V ) -> ( S ` m ) e. _V )
7	4, 5, 6	sylancl 410	. . . . . . 7 |- ( ph -> ( S ` m ) e. _V )
8		frecabex.fvex	. . . . . . 7 |- ( ph -> A. y ( F ` y ) e. _V )
9		fveq2 5486	. . . . . . . . 9 |- ( y = ( S ` m ) -> ( F ` y ) = ( F ` ( S ` m ) ) )
10	9	eleq1d 2235	. . . . . . . 8 |- ( y = ( S ` m ) -> ( ( F ` y ) e. _V <-> ( F ` ( S ` m ) ) e. _V ) )
11	10	spcgv 2813	. . . . . . 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 4121	. . . . . 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 411	. . . . 5 |- ( ph -> { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
15	14	ralrimivw 2540	. . . 4 |- ( ph -> A. m e. om { x | ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
16		abrexex2g 6088	. . . 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 411	. . 3 |- ( ph -> { x | E. m e. om ( dom S = suc m /\ x e. ( F ` ( S ` m ) ) ) } e. _V )
18		simpr 109	. . . . 5 |- ( ( dom S = (/) /\ x e. A ) -> x e. A )
19	18	abssi 3217	. . . 4 |- { x | ( dom S = (/) /\ x e. A ) } C_ A
20		frecabex.aex	. . . 4 |- ( ph -> A e. W )
21		ssexg 4121	. . . 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 411	. . 3 |- ( ph -> { x | ( dom S = (/) /\ x e. A ) } e. _V )
23	17, 22	jca 304	. 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 4420	. . 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 3389	. . . 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 2232	. . 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 183	. 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 121	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 103   \/ wo 698   A.wal 1341   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   _Vcvv 2726   u. cun 3114   C_ wss 3116   (/)c0 3409   suc csuc 4343   omcom 4567   dom cdm 4604   ` cfv 5188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  frectfr  6368