Theorem frectfr 6630

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 6565 or tfri2d 6566, 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 2815	. . . . . . . 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 6628	. . . . . 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 2616	. . . . 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 5484	. . . . 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 2815	. . . 4 |- y e. _V
11		funfvex 5686	. . . . 5 |- ( ( Fun G /\ y e. dom G ) -> ( G ` y ) e. _V )
12	11	funfni 5457	. . . 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 5390	. . 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 2203   {cab 2218   A.wral 2520   E.wrex 2521   _Vcvv 2812   (/)c0 3507   |-> cmpt 4170   suc csuc 4485   omcom 4711   dom cdm 4748   Fun wfun 5345   Fn wfn 5346   ` cfv 5351

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359

This theorem is referenced by:  frecfnom  6631