Theorem freccllem 6397

Description: Lemma for freccl 6398. 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 6386	. . . 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 4899	. . . 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 2201	. . 3 |- frec ( F , A ) = ( G |` om )
5	4	fveq1i 5512	. 2 |- (frec ( F , A ) ` B ) = ( ( G |` om ) ` B )
6		freccl.b	. . . 4 |- ( ph -> B e. om )
7		fvres 5535	. . . 4 |- ( B e. om -> ( ( G |` om ) ` B ) = ( G ` B ) )
8	6, 7	syl 14	. . 3 |- ( ph -> ( ( G |` om ) ` B ) = ( G ` B ) )
9		funmpt 5250	. . . . 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 4603	. . . . 5 |- Ord om
12	11	a1i 9	. . . 4 |- ( ph -> Ord om )
13		vex 2740	. . . . . 6 |- f e. _V
14		simp2 998	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> y e. om )
15		simp3 999	. . . . . . 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 2550	. . . . . . . 8 |- ( ph -> A. z e. S ( F ` z ) e. S )
18	17	3ad2ant1 1018	. . . . . . 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 1018	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> A e. S )
21	14, 15, 18, 20	frecabcl 6394	. . . . . 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 4823	. . . . . . . . . . . 12 |- ( g = f -> dom g = dom f )
23	22	eqeq1d 2186	. . . . . . . . . . 11 |- ( g = f -> ( dom g = suc m <-> dom f = suc m ) )
24		fveq1 5510	. . . . . . . . . . . . 13 |- ( g = f -> ( g ` m ) = ( f ` m ) )
25	24	fveq2d 5515	. . . . . . . . . . . 12 |- ( g = f -> ( F ` ( g ` m ) ) = ( F ` ( f ` m ) ) )
26	25	eleq2d 2247	. . . . . . . . . . 11 |- ( g = f -> ( x e. ( F ` ( g ` m ) ) <-> x e. ( F ` ( f ` m ) ) ) )
27	23, 26	anbi12d 473	. . . . . . . . . 10 |- ( g = f -> ( ( dom g = suc m /\ x e. ( F ` ( g ` m ) ) ) <-> ( dom f = suc m /\ x e. ( F ` ( f ` m ) ) ) ) )
28	27	rexbidv 2478	. . . . . . . . 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 2186	. . . . . . . . . 10 |- ( g = f -> ( dom g = (/) <-> dom f = (/) ) )
30	29	anbi1d 465	. . . . . . . . 9 |- ( g = f -> ( ( dom g = (/) /\ x e. A ) <-> ( dom f = (/) /\ x e. A ) ) )
31	28, 30	orbi12d 793	. . . . . . . 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 2295	. . . . . . 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 2177	. . . . . . 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 5588	. . . . . 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 414	. . . . 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 2254	. . . 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 4610	. . . . . . 7 |- Lim om
38		limuni 4393	. . . . . . 7 |- ( Lim om -> om = U. om )
39	37, 38	ax-mp 5	. . . . . 6 |- om = U. om
40	39	eleq2i 2244	. . . . 5 |- ( y e. om <-> y e. U. om )
41		peano2 4591	. . . . . 6 |- ( y e. om -> suc y e. om )
42	41	adantl 277	. . . . 5 |- ( ( ph /\ y e. om ) -> suc y e. om )
43	40, 42	sylan2br 288	. . . 4 |- ( ( ph /\ y e. U. om ) -> suc y e. om )
44	6, 39	eleqtrdi 2270	. . . 4 |- ( ph -> B e. U. om )
45	2, 10, 12, 36, 43, 44	tfrcl 6359	. . 3 |- ( ph -> ( G ` B ) e. S )
46	8, 45	eqeltrd 2254	. 2 |- ( ph -> ( ( G |` om ) ` B ) e. S )
47	5, 46	eqeltrid 2264	1 |- ( ph -> (frec ( F , A ) ` B ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 708   /\ w3a 978   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737   (/)c0 3422   U.cuni 3807   |-> cmpt 4061   Ord word 4359   Lim wlim 4361   suc csuc 4362   omcom 4586   dom cdm 4623   |` cres 4625   Fun wfun 5206   -->wf 5208   ` cfv 5212  recscrecs 6299  freccfrec 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-recs 6300  df-frec 6386

This theorem is referenced by:  freccl  6398