Theorem frecsuclem2 6012

Description: Lemma for frecsuc 6014. (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 4171	. . . 4 |- ( B e. om -> B e. suc B )
2		fvres 5219	. . . 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 6001	. . . . . 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 5944	. . . . . . . 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 4626	. . . . . 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 2104	. . . . 5 |- frec ( F , A ) = (recs ( G ) |` om )
10	9	fveq1i 5199	. . . 4 |- (frec ( F , A ) ` B ) = ( (recs ( G ) |` om ) ` B )
11		fvres 5219	. . . 4 |- ( B e. om -> ( (recs ( G ) |` om ) ` B ) = (recs ( G ) ` B ) )
12	10, 11	syl5eq 2125	. . 3 |- ( B e. om -> (frec ( F , A ) ` B ) = (recs ( G ) ` B ) )
13	3, 12	eqtr4d 2116	. 2 |- ( B e. om -> ( (recs ( G ) |` suc B ) ` B ) = (frec ( F , A ) ` B ) )
14	13	3ad2ant3 961	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 102   \/ wo 661   /\ w3a 919   A.wal 1282   = wceq 1284   e. wcel 1433   {cab 2067   E.wrex 2349   _Vcvv 2601   (/)c0 3251   |-> cmpt 3839   suc csuc 4120   omcom 4331   dom cdm 4363   |` cres 4365   ` cfv 4922  recscrecs 5942  freccfrec 6000

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-suc 4126  df-xp 4369  df-res 4375  df-iota 4887  df-fv 4930  df-recs 5943  df-frec 6001

This theorem is referenced by:  frecsuclem3  6013