Theorem frecsuclem2 5989

Description: Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.)

Hypothesis

Ref	Expression
frecsuclem1.h	|- 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
frecsuclem2	|- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( (recs ( G ) |` suc B ) ` B ) = (frec ( F , A ) ` B ) )

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

Allowed substitution hint:   V( z)

Proof of Theorem frecsuclem2

Step	Hyp	Ref	Expression
1		sucidg 4153	. . . 4 |- ( B e. om -> B e. suc B )
2		fvres 5198	. . . 4 |- ( B e. suc B -> ( (recs ( G ) |` suc B ) ` B ) = (recs ( G ) ` B ) )
3	1, 2	syl 14	. . 3 |- ( B e. om -> ( (recs ( G ) |` suc B ) ` B ) = (recs ( G ) ` B ) )
4		df-frec 5978	. . . . . 6 |- frec ( F , A ) = (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) |` om )
5		frecsuclem1.h	. . . . . . . 8 |- G = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
6		recseq 5921	. . . . . . . 8 |- ( G = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) -> recs ( G ) = recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) )
7	5, 6	ax-mp 7	. . . . . . 7 |- recs ( G ) = recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) )
8	7	reseq1i 4608	. . . . . 6 |- (recs ( G ) |` om ) = (recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ) |` om )
9	4, 8	eqtr4i 2063	. . . . 5 |- frec ( F , A ) = (recs ( G ) |` om )
10	9	fveq1i 5179	. . . 4 |- (frec ( F , A ) ` B ) = ( (recs ( G ) |` om ) ` B )
11		fvres 5198	. . . 4 |- ( B e. om -> ( (recs ( G ) |` om ) ` B ) = (recs ( G ) ` B ) )
12	10, 11	syl5eq 2084	. . 3 |- ( B e. om -> (frec ( F , A ) ` B ) = (recs ( G ) ` B ) )
13	3, 12	eqtr4d 2075	. 2 |- ( B e. om -> ( (recs ( G ) |` suc B ) ` B ) = (frec ( F , A ) ` B ) )
14	13	3ad2ant3 927	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> ( (recs ( G ) |` suc B ) ` B ) = (frec ( F , A ) ` B ) )

Syntax hints:   -> wi 4   /\ wa 97   \/ wo 629   /\ w3a 885   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   (/)c0 3224   |-> cmpt 3818   suc csuc 4102   omcom 4313   dom cdm 4345   |` cres 4347   ` cfv 4902  recscrecs 5919  freccfrec 5977

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-suc 4108  df-xp 4351  df-res 4357  df-iota 4867  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  frecsuclem3  5990