Theorem frecsuclem1 5987

Description: Lemma for frecsuc 5991. (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 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 )
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 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 ) ) } ) ) )
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 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 )
6	1, 5	eqtr4i 2063	. . . . 5 |- frec ( F , A ) = (recs ( G ) |` om )
7	6	fveq1i 5179	. . . 4 |- (frec ( F , A ) ` suc B ) = ( (recs ( G ) |` om ) ` suc B )
8		peano2 4318	. . . . 5 |- ( B e. om -> suc B e. om )
9		fvres 5198	. . . . 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 2084	. . 3 |- ( B e. om -> (frec ( F , A ) ` suc B ) = (recs ( G ) ` suc B ) )
12	11	3ad2ant3 927	. 2 |- ( ( A. z ( F ` z ) e. _V /\ A e. V /\ B e. om ) -> (frec ( F , A ) ` suc B ) = (recs ( G ) ` suc B ) )
13		nnon 4332	. . . . 5 |- ( suc B e. om -> suc B e. On )
14	8, 13	syl 14	. . . 4 |- ( B e. om -> suc B e. On )
15		eqid 2040	. . . . 5 |- recs ( G ) = recs ( G )
16	2	frectfr 5985	. . . . 5 |- ( ( A. z ( F ` z ) e. _V /\ A e. V ) -> A. y ( Fun G /\ ( G ` y ) e. _V ) )
17	15, 16	tfri2d 5950	. . . 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 270	. . 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 1099	. 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 2072	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 97   \/ wo 629   /\ w3a 885   A.wal 1241   = wceq 1243   e. wcel 1393   {cab 2026   E.wrex 2307   _Vcvv 2557   (/)c0 3224   |-> cmpt 3818   Oncon0 4100   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-in1 544  ax-in2 545  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-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920  df-frec 5978

This theorem is referenced by:  frecsuclem3  5990