Theorem frectfr 6297

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 6232 or tfri2d 6233, 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 2689	. . . . . . . 8 |- g e. _V
2	1	a1i 9	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> g e. _V )
3		simpl 108	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. z ( F ` z ) e. _V )
4		simpr 109	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A e. V )
5	2, 3, 4	frecabex 6295	. . . . . 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 2506	. . . . 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 5249	. . . . 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 2689	. . . 4 |- y e. _V
11		funfvex 5438	. . . . 5 |- ( ( Fun G /\ y e. dom G ) -> ( G ` y ) e. _V )
12	11	funfni 5223	. . . 4 |- ( ( G Fn _V /\ y e. _V ) -> ( G ` y ) e. _V )
13	9, 10, 12	sylancl 409	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( G ` y ) e. _V )
14	7	funmpt2 5162	. . 3 |- Fun G
15	13, 14	jctil 310	. 2 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( Fun G /\ ( G ` y ) e. _V ) )
16	15	alrimiv 1846	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 697   A.wal 1329   = wceq 1331   e. wcel 1480   {cab 2125   A.wral 2416   E.wrex 2417   _Vcvv 2686   (/)c0 3363   |-> cmpt 3989   suc csuc 4287   omcom 4504   dom cdm 4539   Fun wfun 5117   Fn wfn 5118   ` cfv 5123

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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131

This theorem is referenced by:  frecfnom  6298