Theorem frectfr 6368

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 6303 or tfri2d 6304, 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 2729	. . . . . . . 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 6366	. . . . . 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 2540	. . . . 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 5314	. . . . 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 2729	. . . 4 |- y e. _V
11		funfvex 5503	. . . . 5 |- ( ( Fun G /\ y e. dom G ) -> ( G ` y ) e. _V )
12	11	funfni 5288	. . . 4 |- ( ( G Fn _V /\ y e. _V ) -> ( G ` y ) e. _V )
13	9, 10, 12	sylancl 410	. . 3 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> ( G ` y ) e. _V )
14	7	funmpt2 5227	. . 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 1862	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 698   A.wal 1341   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   _Vcvv 2726   (/)c0 3409   |-> cmpt 4043   suc csuc 4343   omcom 4567   dom cdm 4604   Fun wfun 5182   Fn wfn 5183   ` cfv 5188

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196

This theorem is referenced by:  frecfnom  6369