Theorem freccllem 6346

Description: Lemma for freccl 6347. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)

Hypotheses

Ref	Expression
freccl.a	|- ( ph -> A e. S )
freccl.cl	|- ( ( ph /\ z e. S ) -> ( F ` z ) e. S )
freccl.b	|- ( ph -> B e. om )
freccllem.g	|- G = recs ( ( 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
freccllem	|- ( ph -> (frec ( F , A ) ` B ) e. S )

Distinct variable groups:   A, g, m, x   z, A, m, x   x, B   g, F, m, x   z, F   S, m, x, z   ph, m, x, z

Allowed substitution hints:   ph( g)   B( z, g, m)   S( g)   G( x, z, g, m)

Proof of Theorem freccllem

Dummy variables f y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-frec 6335	. . . 4 |- 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		freccllem.g	. . . . 5 |- G = recs ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) )
3	2	reseq1i 4861	. . . 4 |- ( 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 )
4	1, 3	eqtr4i 2181	. . 3 |- frec ( F , A ) = ( G |` om )
5	4	fveq1i 5468	. 2 |- (frec ( F , A ) ` B ) = ( ( G |` om ) ` B )
6		freccl.b	. . . 4 |- ( ph -> B e. om )
7		fvres 5491	. . . 4 |- ( B e. om -> ( ( G |` om ) ` B ) = ( G ` B ) )
8	6, 7	syl 14	. . 3 |- ( ph -> ( ( G |` om ) ` B ) = ( G ` B ) )
9		funmpt 5207	. . . . 5 |- Fun ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
10	9	a1i 9	. . . 4 |- ( ph -> Fun ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) )
11		ordom 4565	. . . . 5 |- Ord om
12	11	a1i 9	. . . 4 |- ( ph -> Ord om )
13		vex 2715	. . . . . 6 |- f e. _V
14		simp2 983	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> y e. om )
15		simp3 984	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> f : y --> S )
16		freccl.cl	. . . . . . . . 9 |- ( ( ph /\ z e. S ) -> ( F ` z ) e. S )
17	16	ralrimiva 2530	. . . . . . . 8 |- ( ph -> A. z e. S ( F ` z ) e. S )
18	17	3ad2ant1 1003	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> A. z e. S ( F ` z ) e. S )
19		freccl.a	. . . . . . . 8 |- ( ph -> A e. S )
20	19	3ad2ant1 1003	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> A e. S )
21	14, 15, 18, 20	frecabcl 6343	. . . . . 6 |- ( ( ph /\ y e. om /\ f : y --> S ) -> { x | ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } e. S )
22		dmeq 4785	. . . . . . . . . . . 12 |- ( g = f -> dom g = dom f )
23	22	eqeq1d 2166	. . . . . . . . . . 11 |- ( g = f -> ( dom g = suc m <-> dom f = suc m ) )
24		fveq1 5466	. . . . . . . . . . . . 13 |- ( g = f -> ( g ` m ) = ( f ` m ) )
25	24	fveq2d 5471	. . . . . . . . . . . 12 |- ( g = f -> ( F ` ( g ` m ) ) = ( F ` ( f ` m ) ) )
26	25	eleq2d 2227	. . . . . . . . . . 11 |- ( g = f -> ( x e. ( F ` ( g ` m ) ) <-> x e. ( F ` ( f ` m ) ) ) )
27	23, 26	anbi12d 465	. . . . . . . . . 10 |- ( g = f -> ( ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) <-> ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) ) )
28	27	rexbidv 2458	. . . . . . . . 9 |- ( g = f -> ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) <-> E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) ) )
29	22	eqeq1d 2166	. . . . . . . . . 10 |- ( g = f -> ( dom g = (/) <-> dom f = (/) ) )
30	29	anbi1d 461	. . . . . . . . 9 |- ( g = f -> ( ( dom g = (/) /\ x e. A ) <-> ( dom f = (/) /\ x e. A ) ) )
31	28, 30	orbi12d 783	. . . . . . . 8 |- ( g = f -> ( ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) <-> ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) ) )
32	31	abbidv 2275	. . . . . . 7 |- ( g = f -> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } = { x | ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } )
33		eqid 2157	. . . . . . 7 |- ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) = ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } )
34	32, 33	fvmptg 5543	. . . . . 6 |- ( ( f e. _V /\ { x | ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } e. S ) -> ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ` f ) = { x | ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } )
35	13, 21, 34	sylancr 411	. . . . 5 |- ( ( ph /\ y e. om /\ f : y --> S ) -> ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ` f ) = { x | ( E. m e. om ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) \/ ( dom f = (/) /\ x e. A ) ) } )
36	35, 21	eqeltrd 2234	. . . 4 |- ( ( ph /\ y e. om /\ f : y --> S ) -> ( ( g e. _V |-> { x | ( E. m e. om ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) \/ ( dom g = (/) /\ x e. A ) ) } ) ` f ) e. S )
37		limom 4572	. . . . . . 7 |- Lim om
38		limuni 4356	. . . . . . 7 |- ( Lim om -> om = U. om )
39	37, 38	ax-mp 5	. . . . . 6 |- om = U. om
40	39	eleq2i 2224	. . . . 5 |- ( y e. om <-> y e. U. om )
41		peano2 4553	. . . . . 6 |- ( y e. om -> suc y e. om )
42	41	adantl 275	. . . . 5 |- ( ( ph /\ y e. om ) -> suc y e. om )
43	40, 42	sylan2br 286	. . . 4 |- ( ( ph /\ y e. U. om ) -> suc y e. om )
44	6, 39	eleqtrdi 2250	. . . 4 |- ( ph -> B e. U. om )
45	2, 10, 12, 36, 43, 44	tfrcl 6308	. . 3 |- ( ph -> ( G ` B ) e. S )
46	8, 45	eqeltrd 2234	. 2 |- ( ph -> ( ( G |` om ) ` B ) e. S )
47	5, 46	eqeltrid 2244	1 |- ( ph -> (frec ( F , A ) ` B ) e. S )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   /\ w3a 963   = wceq 1335   e. wcel 2128   {cab 2143   A.wral 2435   E.wrex 2436   _Vcvv 2712   (/)c0 3394   U.cuni 3772   |-> cmpt 4025   Ord word 4322   Lim wlim 4324   suc csuc 4325   omcom 4548   dom cdm 4585   |` cres 4587   Fun wfun 5163   -->wf 5165   ` cfv 5169  recscrecs 6248  freccfrec 6334

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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-iinf 4546

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-ilim 4329  df-suc 4331  df-iom 4549  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-recs 6249  df-frec 6335

This theorem is referenced by:  freccl  6347