Theorem frecsuc 5991

Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 15-Aug-2019.)

Assertion

Ref	Expression
frecsuc	|- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( F ` (frec ( F , A ) ` B ) ) )

Distinct variable groups:   z, A   z, B   z, F

Allowed substitution hint:   V( z)

Proof of Theorem frecsuc

Dummy variables f g m x y n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		suceq 4139	. . . . . . . . . 10 |- ( n = m -> suc n = suc m )
2	1	eqeq2d 2051	. . . . . . . . 9 |- ( n = m -> ( dom f = suc n <-> dom f = suc m ) )
3		fveq2 5178	. . . . . . . . . . 11 |- ( n = m -> ( f ` n ) = ( f ` m ) )
4	3	fveq2d 5182	. . . . . . . . . 10 |- ( n = m -> ( F ` ( f ` n ) ) = ( F ` ( f ` m ) ) )
5	4	eleq2d 2107	. . . . . . . . 9 |- ( n = m -> ( x e. ( F ` ( f ` n ) ) <-> x e. ( F ` ( f ` m ) ) ) )
6	2, 5	anbi12d 442	. . . . . . . 8 |- ( n = m -> ( ( dom f = suc n /\ x e. ( F ` ( f ` n ) ) ) <-> ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) ) )
7	6	cbvrexv 2534	. . . . . . 7 |- ( E. n e. om ( dom f = suc n /\ x e. ( F ` ( f ` n ) ) ) <-> E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) )
8	7	orbi1i 680	. . . . . 6 |- ( ( E. n e. om ( dom f = suc n /\ x e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ x e. A ) ) <-> ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) )
9	8	abbii 2153	. . . . 5 |- { x | ( E. n e. om ( dom f = suc n /\ x e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } = { x | ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) }
10		eleq1 2100	. . . . . . . . 9 |- ( x = y -> ( x e. ( F ` ( f ` m ) ) <-> y e. ( F ` ( f ` m ) ) ) )
11	10	anbi2d 437	. . . . . . . 8 |- ( x = y -> ( ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) <-> ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) ) )
12	11	rexbidv 2327	. . . . . . 7 |- ( x = y -> ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) <-> E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) ) )
13		eleq1 2100	. . . . . . . 8 |- ( x = y -> ( x e. A <-> y e. A ) )
14	13	anbi2d 437	. . . . . . 7 |- ( x = y -> ( ( dom f = (/) /\ x e. A ) <-> ( dom f = (/) /\ y e. A ) ) )
15	12, 14	orbi12d 707	. . . . . 6 |- ( x = y -> ( ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) <-> ( E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ y e. A ) ) ) )
16	15	cbvabv 2161	. . . . 5 |- { x | ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } = { y | ( E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ y e. A ) ) }
17	9, 16	eqtri 2060	. . . 4 |- { x | ( E. n e. om ( dom f = suc n /\ x e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } = { y | ( E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ y e. A ) ) }
18	17	mpteq2i 3844	. . 3 |- ( f e. _V |-> { x | ( E. n e. om ( dom f = suc n /\ x e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } ) = ( f e. _V |-> { y | ( E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ y e. A ) ) } )
19		dmeq 4535	. . . . . . . . 9 |- ( f = g -> dom f = dom g )
20	19	eqeq1d 2048	. . . . . . . 8 |- ( f = g -> ( dom f = suc m <-> dom g = suc m ) )
21		fveq1 5177	. . . . . . . . . 10 |- ( f = g -> ( f ` m ) = ( g ` m ) )
22	21	fveq2d 5182	. . . . . . . . 9 |- ( f = g -> ( F ` ( f ` m ) ) = ( F ` ( g ` m ) ) )
23	22	eleq2d 2107	. . . . . . . 8 |- ( f = g -> ( y e. ( F ` ( f ` m ) ) <-> y e. ( F ` ( g ` m ) ) ) )
24	20, 23	anbi12d 442	. . . . . . 7 |- ( f = g -> ( ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) <-> ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) ) )
25	24	rexbidv 2327	. . . . . 6 |- ( f = g -> ( E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) <-> E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) ) )
26	19	eqeq1d 2048	. . . . . . 7 |- ( f = g -> ( dom f = (/) <-> dom g = (/) ) )
27	26	anbi1d 438	. . . . . 6 |- ( f = g -> ( ( dom f = (/) /\ y e. A ) <-> ( dom g = (/) /\ y e. A ) ) )
28	25, 27	orbi12d 707	. . . . 5 |- ( f = g -> ( ( E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ y e. A ) ) <-> ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) ) )
29	28	abbidv 2155	. . . 4 |- ( f = g -> { y | ( E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ y e. A ) ) } = { y | ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } )
30	29	cbvmptv 3852	. . 3 |- ( f e. _V |-> { y | ( E. m e. om ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ y e. A ) ) } ) = ( g e. _V |-> { y | ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } )
31	18, 30	eqtri 2060	. 2 |- ( f e. _V |-> { x | ( E. n e. om ( dom f = suc n /\ x e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } ) = ( g e. _V |-> { y | ( E. m e. om ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } )
32	31	frecsuclem3 5990	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( F ` (frec ( F , A ) ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   /\ w3a 885   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   (/)c0 3224   |-> cmpt 3818   suc csuc 4102   omcom 4313   dom cdm 4345   ` cfv 4902  freccfrec 5977

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  frecrdg  5992  freccl  5993  frec2uzzd  9186  frec2uzsucd  9187  frec2uzrdg  9195  frecuzrdgsuc  9201