Theorem frectfr 6644

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 6579 or tfri2d 6580, 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 2818	. . . . . . . 8 |- g e. _V
2	1	a1i 9	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> g e. _V )
3		simpl 109	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. z ( F ` z ) e. _V )
4		simpr 110	. . . . . . 7 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A e. V )
5	2, 3, 4	frecabex 6642	. . . . . 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 2618	. . . . 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 5490	. . . . 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 2818	. . . 4 |- y e. _V
11		funfvex 5692	. . . . 5 |- ( ( Fun G /\ y e. dom G ) -> ( G ` y ) e. _V )
12	11	funfni 5463	. . . 4 |- ( ( G Fn _V /\ y e. _V ) -> ( G ` y ) e. _V )
13	9, 10, 12	sylancl 413	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( G ` y ) e. _V )
14	7	funmpt2 5396	. . 3 |- Fun G
15	13, 14	jctil 312	. 2 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( Fun G /\ ( G ` y ) e. _V ) )
16	15	alrimiv 1923	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   A.wal 1396   = wceq 1398   e. wcel 2205   {cab 2220   A.wral 2522   E.wrex 2523   _Vcvv 2815   (/)c0 3512   |-> cmpt 4176   suc csuc 4491   omcom 4717   dom cdm 4754   Fun wfun 5351   Fn wfn 5352   ` cfv 5357

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365

This theorem is referenced by:  frecfnom  6645