Theorem frectfr 6008

Description: Lemma to connect transfinite recursion theorems with finite recursion. That is, given the conditions F Fn _V and A e. V on frec ( F , A ), we want to be able to apply tfri1d 5972 or tfri2d 5973, and this lemma lets us satisfy hypotheses of those theorems.

(Contributed by Jim Kingdon, 15-Aug-2019.)

Hypothesis

Ref	Expression
frectfr.1	|- G = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )

Assertion

Ref	Expression
frectfr	|- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )

Distinct variable groups:   g, m, x, y, A   z, g, F, m, x, y   g, V, m, y

Allowed substitution hints:   A( z)   G( x, y, z, g, m)   V( x, z)

Proof of Theorem frectfr

Step	Hyp	Ref	Expression
1		vex 2604	. . . . . . . 8 |- g e. _V
2	1	a1i 9	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> g e. _V )
3		simpl 107	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. z ( F ` z ) e. _V )
4		simpr 108	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A e. V )
5	2, 3, 4	frecabex 6007	. . . . . 6 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } e. _V )
6	5	ralrimivw 2435	. . . . 5 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. g e. _V { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } e. _V )
7		frectfr.1	. . . . . 6 |- G = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
8	7	fnmpt 5045	. . . . 5 |- ( A. g e. _V { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } e. _V -> G Fn _V )
9	6, 8	syl 14	. . . 4 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> G Fn _V )
10		vex 2604	. . . 4 |- y e. _V
11		funfvex 5212	. . . . 5 |- ( ( Fun G /\ y e. dom G ) -> ( G ` y ) e. _V )
12	11	funfni 5019	. . . 4 |- ( ( G Fn _V /\ y e. _V ) -> ( G ` y ) e. _V )
13	9, 10, 12	sylancl 404	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( G ` y ) e. _V )
14	7	funmpt2 4959	. . 3 |- Fun G
15	13, 14	jctil 305	. 2 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( Fun G /\ ( G ` y ) e. _V ) )
16	15	alrimiv 1795	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   A.wal 1282   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   _Vcvv 2601   (/)c0 3251   |-> cmpt 3839   suc csuc 4120   omcom 4331   dom cdm 4363   Fun wfun 4916   Fn wfn 4917   ` cfv 4922

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-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-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  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-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  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-id 4048  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

This theorem is referenced by:  frecfnom  6009  frecsuclem1  6010  frecsuclemdm  6011  frecsuclem3  6013