Theorem frecsuc 6375

Description: The successor value resulting from finite recursive definition generation. (Contributed by Jim Kingdon, 31-Mar-2022.)

Assertion

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

Distinct variable groups:   z, F   z, S

Allowed substitution hints:   A( z)   B( 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		dmeq 4804	. . . . . . . . 9 |- ( f = g -> dom f = dom g )
2	1	eqeq1d 2174	. . . . . . . 8 |- ( f = g -> ( dom f = suc n <-> dom g = suc n ) )
3		fveq1 5485	. . . . . . . . . 10 |- ( f = g -> ( f ` n ) = ( g ` n ) )
4	3	fveq2d 5490	. . . . . . . . 9 |- ( f = g -> ( F ` ( f ` n ) ) = ( F ` ( g ` n ) ) )
5	4	eleq2d 2236	. . . . . . . 8 |- ( f = g -> ( y e. ( F ` ( f ` n ) ) <-> y e. ( F ` ( g ` n ) ) ) )
6	2, 5	anbi12d 465	. . . . . . 7 |- ( f = g -> ( ( dom f = suc n /\ y e. ( F ` ( f ` n ) ) ) <-> ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) ) )
7	6	rexbidv 2467	. . . . . 6 |- ( f = g -> ( E. n e. om ( dom f = suc n /\ y e. ( F ` ( f ` n ) ) ) <-> E. n e. om ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) ) )
8	1	eqeq1d 2174	. . . . . . 7 |- ( f = g -> ( dom f = (/) <-> dom g = (/) ) )
9	8	anbi1d 461	. . . . . 6 |- ( f = g -> ( ( dom f = (/) /\ y e. A ) <-> ( dom g = (/) /\ y e. A ) ) )
10	7, 9	orbi12d 783	. . . . 5 |- ( f = g -> ( ( E. n e. om ( dom f = suc n /\ y e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ y e. A ) ) <-> ( E. n e. om ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ y e. A ) ) ) )
11	10	abbidv 2284	. . . 4 |- ( f = g -> { y | ( E. n e. om ( dom f = suc n /\ y e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ y e. A ) ) } = { y | ( E. n e. om ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } )
12	11	cbvmptv 4078	. . 3 |- ( f e. _V |-> { y | ( E. n e. om ( dom f = suc n /\ y e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ y e. A ) ) } ) = ( g e. _V |-> { y | ( E. n e. om ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } )
13		eleq1 2229	. . . . . . . 8 |- ( y = x -> ( y e. ( F ` ( g ` n ) ) <-> x e. ( F ` ( g ` n ) ) ) )
14	13	anbi2d 460	. . . . . . 7 |- ( y = x -> ( ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) <-> ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) ) )
15	14	rexbidv 2467	. . . . . 6 |- ( y = x -> ( E. n e. om ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) <-> E. n e. om ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) ) )
16		eleq1 2229	. . . . . . 7 |- ( y = x -> ( y e. A <-> x e. A ) )
17	16	anbi2d 460	. . . . . 6 |- ( y = x -> ( ( dom g = (/) /\ y e. A ) <-> ( dom g = (/) /\ x e. A ) ) )
18	15, 17	orbi12d 783	. . . . 5 |- ( y = x -> ( ( E. n e. om ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ y e. A ) ) <-> ( E. n e. om ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ x e. A ) ) ) )
19	18	cbvabv 2291	. . . 4 |- { y | ( E. n e. om ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } = { x | ( E. n e. om ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ x e. A ) ) }
20	19	mpteq2i 4069	. . 3 |- ( g e. _V |-> { y | ( E. n e. om ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ y e. A ) ) } ) = ( g e. _V |-> { x | ( E. n e. om ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
21		suceq 4380	. . . . . . . . 9 |- ( n = m -> suc n = suc m )
22	21	eqeq2d 2177	. . . . . . . 8 |- ( n = m -> ( dom g = suc n <-> dom g = suc m ) )
23		fveq2 5486	. . . . . . . . . 10 |- ( n = m -> ( g ` n ) = ( g ` m ) )
24	23	fveq2d 5490	. . . . . . . . 9 |- ( n = m -> ( F ` ( g ` n ) ) = ( F ` ( g ` m ) ) )
25	24	eleq2d 2236	. . . . . . . 8 |- ( n = m -> ( x e. ( F ` ( g ` n ) ) <-> x e. ( F ` ( g ` m ) ) ) )
26	22, 25	anbi12d 465	. . . . . . 7 |- ( n = m -> ( ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) <-> ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) ) )
27	26	cbvrexv 2693	. . . . . 6 |- ( E. n e. om ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) <-> E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) )
28	27	orbi1i 753	. . . . 5 |- ( ( E. n e. om ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ x e. A ) ) <-> ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) )
29	28	abbii 2282	. . . 4 |- { x | ( E. n e. om ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } = { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) }
30	29	mpteq2i 4069	. . 3 |- ( g e. _V |-> { x | ( E. n e. om ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
31	12, 20, 30	3eqtri 2190	. 2 |- ( f e. _V |-> { y | ( E. n e. om ( dom f = suc n /\ y e. ( F ` ( f ` n ) ) ) \/ ( dom f = (/) /\ y e. A ) ) } ) = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
32	31	frecsuclem 6374	1 |- ( ( A. z e. S ( F ` z ) e. S /\ A e. S /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( F ` (frec ( F , A ) ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   /\ w3a 968   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   _Vcvv 2726   (/)c0 3409   |-> cmpt 4043   suc csuc 4343   omcom 4567   dom cdm 4604   ` cfv 5188  freccfrec 6358

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-in1 604  ax-in2 605  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-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  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-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  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-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  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  df-recs 6273  df-frec 6359

This theorem is referenced by:  frecrdg  6376  frec2uzsucd  10336  frec2uzrdg  10344  frecuzrdgsuc  10349  frecuzrdgg  10351  frecuzrdgsuctlem  10358  seq3val  10393  seqvalcd  10394