Theorem frecsuc 6014

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 4157	. . . . . . . . . 10 |- ( n = m -> suc n = suc m )
2	1	eqeq2d 2092	. . . . . . . . 9 |- ( n = m -> ( dom f = suc n <-> dom f = suc m ) )
3		fveq2 5198	. . . . . . . . . . 11 |- ( n = m -> ( f ` n ) = ( f ` m ) )
4	3	fveq2d 5202	. . . . . . . . . 10 |- ( n = m -> ( F ` ( f ` n ) ) = ( F ` ( f ` m ) ) )
5	4	eleq2d 2148	. . . . . . . . 9 |- ( n = m -> ( x e. ( F ` ( f ` n ) ) <-> x e. ( F ` ( f ` m ) ) ) )
6	2, 5	anbi12d 456	. . . . . . . 8 |- ( n = m -> ( ( dom f = suc n /\ x e. ( F ` ( f ` n ) ) ) <-> ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) ) )
7	6	cbvrexv 2578	. . . . . . 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 712	. . . . . 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 2194	. . . . 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 2141	. . . . . . . . 9 |- ( x = y -> ( x e. ( F ` ( f ` m ) ) <-> y e. ( F ` ( f ` m ) ) ) )
11	10	anbi2d 451	. . . . . . . 8 |- ( x = y -> ( ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) <-> ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) ) )
12	11	rexbidv 2369	. . . . . . 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 2141	. . . . . . . 8 |- ( x = y -> ( x e. A <-> y e. A ) )
14	13	anbi2d 451	. . . . . . 7 |- ( x = y -> ( ( dom f = (/) /\ x e. A ) <-> ( dom f = (/) /\ y e. A ) ) )
15	12, 14	orbi12d 739	. . . . . 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 2202	. . . . 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 2101	. . . 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 3865	. . 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 4553	. . . . . . . . 9 |- ( f = g -> dom f = dom g )
20	19	eqeq1d 2089	. . . . . . . 8 |- ( f = g -> ( dom f = suc m <-> dom g = suc m ) )
21		fveq1 5197	. . . . . . . . . 10 |- ( f = g -> ( f ` m ) = ( g ` m ) )
22	21	fveq2d 5202	. . . . . . . . 9 |- ( f = g -> ( F ` ( f ` m ) ) = ( F ` ( g ` m ) ) )
23	22	eleq2d 2148	. . . . . . . 8 |- ( f = g -> ( y e. ( F ` ( f ` m ) ) <-> y e. ( F ` ( g ` m ) ) ) )
24	20, 23	anbi12d 456	. . . . . . 7 |- ( f = g -> ( ( dom f = suc m /\ y e. ( F ` ( f ` m ) ) ) <-> ( dom g = suc m /\ y e. ( F ` ( g ` m ) ) ) ) )
25	24	rexbidv 2369	. . . . . 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 2089	. . . . . . 7 |- ( f = g -> ( dom f = (/) <-> dom g = (/) ) )
27	26	anbi1d 452	. . . . . 6 |- ( f = g -> ( ( dom f = (/) /\ y e. A ) <-> ( dom g = (/) /\ y e. A ) ) )
28	25, 27	orbi12d 739	. . . . 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 2196	. . . 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 3873	. . 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 2101	. 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 6013	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 102   \/ wo 661   /\ w3a 919   A.wal 1282   = wceq 1284   e. wcel 1433   {cab 2067   E.wrex 2349   _Vcvv 2601   (/)c0 3251   |-> cmpt 3839   suc csuc 4120   omcom 4331   dom cdm 4363   ` cfv 4922  freccfrec 6000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-frec 6001

This theorem is referenced by:  frecrdg  6015  freccl  6016  frec2uzzd  9402  frec2uzsucd  9403  frec2uzrdg  9411  frecuzrdgsuc  9417