Theorem frecsuclem1 6010

Description: Lemma for frecsuc 6014. (Contributed by Jim Kingdon, 13-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
frecsuclem1	|- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( G ` (recs ( G ) |` suc 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 frecsuclem1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		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 )
2		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 ) ) } )
3		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 ) ) } ) ) )
4	2, 3	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 ) ) } ) )
5	4	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 )
6	1, 5	eqtr4i 2104	. . . . 5 |- frec ( F , A ) = (recs ( G ) |` om )
7	6	fveq1i 5199	. . . 4 |- (frec ( F , A ) ` suc B ) = ( (recs ( G ) |` om ) ` suc B )
8		peano2 4336	. . . . 5 |- ( B e. om -> suc B e. om )
9		fvres 5219	. . . . 5 |- ( suc B e. om -> ( (recs ( G ) |` om ) ` suc B ) = (recs ( G ) ` suc B ) )
10	8, 9	syl 14	. . . 4 |- ( B e. om -> ( (recs ( G ) |` om ) ` suc B ) = (recs ( G ) ` suc B ) )
11	7, 10	syl5eq 2125	. . 3 |- ( B e. om -> (frec ( F , A ) ` suc B ) = (recs ( G ) ` suc B ) )
12	11	3ad2ant3 961	. 2 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = (recs ( G ) ` suc B ) )
13		nnon 4350	. . . . 5 |- ( suc B e. om -> suc B e. On )
14	8, 13	syl 14	. . . 4 |- ( B e. om -> suc B e. On )
15		eqid 2081	. . . . 5 |- recs ( G ) = recs ( G )
16	2	frectfr 6008	. . . . 5 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )
17	15, 16	tfri2d 5973	. . . 4 |- ( ( ( A. z ( F ` z ) e. _V /\ A e. V ) /\ suc B e. On ) -> (recs ( G ) ` suc B ) = ( G ` (recs ( G ) |` suc B ) ) )
18	14, 17	sylan2 280	. . 3 |- ( ( ( A. z ( F ` z ) e. _V /\ A e. V ) /\ B e. om ) -> (recs ( G ) ` suc B ) = ( G ` (recs ( G ) |` suc B ) ) )
19	18	3impa 1133	. 2 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (recs ( G ) ` suc B ) = ( G ` (recs ( G ) |` suc B ) ) )
20	12, 19	eqtrd 2113	1 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = ( G ` (recs ( G ) |` suc 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   Oncon0 4118   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-in1 576  ax-in2 577  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-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943  df-frec 6001

This theorem is referenced by:  frecsuclem3  6013