Theorem frecsuc 6460

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 4862	. . . . . . . . 9 |- ( f = g -> dom f = dom g )
2	1	eqeq1d 2202	. . . . . . . 8 |- ( f = g -> ( dom f = suc n <-> dom g = suc n ) )
3		fveq1 5553	. . . . . . . . . 10 |- ( f = g -> ( f ` n ) = ( g ` n ) )
4	3	fveq2d 5558	. . . . . . . . 9 |- ( f = g -> ( F ` ( f ` n ) ) = ( F ` ( g ` n ) ) )
5	4	eleq2d 2263	. . . . . . . 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 2495	. . . . . 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 2202	. . . . . . 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 794	. . . . 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 2311	. . . 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 4125	. . 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 2256	. . . . . . . 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 2495	. . . . . 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 2256	. . . . . . 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 794	. . . . 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 2318	. . . 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 4116	. . 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 4433	. . . . . . . . 9 |- ( n = m -> suc n = suc m )
22	21	eqeq2d 2205	. . . . . . . 8 |- ( n = m -> ( dom g = suc n <-> dom g = suc m ) )
23		fveq2 5554	. . . . . . . . . 10 |- ( n = m -> ( g ` n ) = ( g ` m ) )
24	23	fveq2d 5558	. . . . . . . . 9 |- ( n = m -> ( F ` ( g ` n ) ) = ( F ` ( g ` m ) ) )
25	24	eleq2d 2263	. . . . . . . 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 2727	. . . . . 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 764	. . . . 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 2309	. . . 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 4116	. . 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 2218	. 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 6459	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 709   /\ w3a 980   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   _Vcvv 2760   (/)c0 3446   |-> cmpt 4090   suc csuc 4396   omcom 4622   dom cdm 4659   ` cfv 5254  freccfrec 6443

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-ilim 4400  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-recs 6358  df-frec 6444

This theorem is referenced by:  frecrdg  6461  frec2uzsucd  10472  frec2uzrdg  10480  frecuzrdgsuc  10485  frecuzrdgg  10487  frecuzrdgsuctlem  10494  seq3val  10531  seqvalcd  10532