Theorem frecsuc 6516

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 4897	. . . . . . . . 9 |- ( f = g -> dom f = dom g )
2	1	eqeq1d 2216	. . . . . . . 8 |- ( f = g -> ( dom f = suc n <-> dom g = suc n ) )
3		fveq1 5598	. . . . . . . . . 10 |- ( f = g -> ( f ` n ) = ( g ` n ) )
4	3	fveq2d 5603	. . . . . . . . 9 |- ( f = g -> ( F ` ( f ` n ) ) = ( F ` ( g ` n ) ) )
5	4	eleq2d 2277	. . . . . . . 8 |- ( f = g -> ( y e. ( F ` ( f ` n ) ) <-> y e. ( F ` ( g ` n ) ) ) )
6	2, 5	anbi12d 473	. . . . . . 7 |- ( f = g -> ( ( dom f = suc n /\ y e. ( F ` ( f ` n ) ) ) <-> ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) ) )
7	6	rexbidv 2509	. . . . . 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 2216	. . . . . . 7 |- ( f = g -> ( dom f = (/) <-> dom g = (/) ) )
9	8	anbi1d 465	. . . . . 6 |- ( f = g -> ( ( dom f = (/) /\ y e. A ) <-> ( dom g = (/) /\ y e. A ) ) )
10	7, 9	orbi12d 795	. . . . 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 2325	. . . 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 4156	. . 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 2270	. . . . . . . 8 |- ( y = x -> ( y e. ( F ` ( g ` n ) ) <-> x e. ( F ` ( g ` n ) ) ) )
14	13	anbi2d 464	. . . . . . 7 |- ( y = x -> ( ( dom g = suc n /\ y e. ( F ` ( g ` n ) ) ) <-> ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) ) )
15	14	rexbidv 2509	. . . . . 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 2270	. . . . . . 7 |- ( y = x -> ( y e. A <-> x e. A ) )
17	16	anbi2d 464	. . . . . 6 |- ( y = x -> ( ( dom g = (/) /\ y e. A ) <-> ( dom g = (/) /\ x e. A ) ) )
18	15, 17	orbi12d 795	. . . . 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 2332	. . . 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 4147	. . 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 4467	. . . . . . . . 9 |- ( n = m -> suc n = suc m )
22	21	eqeq2d 2219	. . . . . . . 8 |- ( n = m -> ( dom g = suc n <-> dom g = suc m ) )
23		fveq2 5599	. . . . . . . . . 10 |- ( n = m -> ( g ` n ) = ( g ` m ) )
24	23	fveq2d 5603	. . . . . . . . 9 |- ( n = m -> ( F ` ( g ` n ) ) = ( F ` ( g ` m ) ) )
25	24	eleq2d 2277	. . . . . . . 8 |- ( n = m -> ( x e. ( F ` ( g ` n ) ) <-> x e. ( F ` ( g ` m ) ) ) )
26	22, 25	anbi12d 473	. . . . . . 7 |- ( n = m -> ( ( dom g = suc n /\ x e. ( F ` ( g ` n ) ) ) <-> ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) ) )
27	26	cbvrexv 2743	. . . . . 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 765	. . . . 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 2323	. . . 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 4147	. . 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 2232	. 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 6515	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 104   \/ wo 710   /\ w3a 981   = wceq 1373   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   _Vcvv 2776   (/)c0 3468   |-> cmpt 4121   suc csuc 4430   omcom 4656   dom cdm 4693   ` cfv 5290  freccfrec 6499

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-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-recs 6414  df-frec 6500

This theorem is referenced by:  frecrdg  6517  frec2uzsucd  10583  frec2uzrdg  10591  frecuzrdgsuc  10596  frecuzrdgg  10598  frecuzrdgsuctlem  10605  seq3val  10642  seqvalcd  10643