Theorem freccllem 6416

Description: Lemma for freccl 6417. 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 6405	. . . 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 4915	. . . 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 2211	. . 3 |- frec ( F , A ) = ( G |` om )
5	4	fveq1i 5528	. 2 |- (frec ( F , A ) ` B ) = ( ( G |` om ) ` B )
6		freccl.b	. . . 4 |- ( ph -> B e. om )
7		fvres 5551	. . . 4 |- ( B e. om -> ( ( G |` om ) ` B ) = ( G ` B ) )
8	6, 7	syl 14	. . 3 |- ( ph -> ( ( G |` om ) ` B ) = ( G ` B ) )
9		funmpt 5266	. . . . 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 4618	. . . . 5 |- Ord om
12	11	a1i 9	. . . 4 |- ( ph -> Ord om )
13		vex 2752	. . . . . 6 |- f e. _V
14		simp2 999	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> y e. om )
15		simp3 1000	. . . . . . 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 2560	. . . . . . . 8 |- ( ph -> A. z e. S ( F ` z ) e. S )
18	17	3ad2ant1 1019	. . . . . . 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 1019	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> A e. S )
21	14, 15, 18, 20	frecabcl 6413	. . . . . 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 4839	. . . . . . . . . . . 12 |- ( g = f -> dom g = dom f )
23	22	eqeq1d 2196	. . . . . . . . . . 11 |- ( g = f -> ( dom g = suc m <-> dom f = suc m ) )
24		fveq1 5526	. . . . . . . . . . . . 13 |- ( g = f -> ( g ` m ) = ( f ` m ) )
25	24	fveq2d 5531	. . . . . . . . . . . 12 |- ( g = f -> ( F ` ( g ` m ) ) = ( F ` ( f ` m ) ) )
26	25	eleq2d 2257	. . . . . . . . . . 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 2488	. . . . . . . . 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 2196	. . . . . . . . . 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 794	. . . . . . . 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 2305	. . . . . . 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 2187	. . . . . . 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 5605	. . . . . 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 2264	. . . 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 4625	. . . . . . 7 |- Lim om
38		limuni 4408	. . . . . . 7 |- ( Lim om -> om = U. om )
39	37, 38	ax-mp 5	. . . . . 6 |- om = U. om
40	39	eleq2i 2254	. . . . 5 |- ( y e. om <-> y e. U. om )
41		peano2 4606	. . . . . 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 2280	. . . 4 |- ( ph -> B e. U. om )
45	2, 10, 12, 36, 43, 44	tfrcl 6378	. . 3 |- ( ph -> ( G ` B ) e. S )
46	8, 45	eqeltrd 2264	. 2 |- ( ph -> ( ( G |` om ) ` B ) e. S )
47	5, 46	eqeltrid 2274	1 |- ( ph -> (frec ( F , A ) ` B ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   /\ w3a 979   = wceq 1363   e. wcel 2158   {cab 2173   A.wral 2465   E.wrex 2466   _Vcvv 2749   (/)c0 3434   U.cuni 3821   |-> cmpt 4076   Ord word 4374   Lim wlim 4376   suc csuc 4377   omcom 4601   dom cdm 4638   |` cres 4640   Fun wfun 5222   -->wf 5224   ` cfv 5228  recscrecs 6318  freccfrec 6404

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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-recs 6319  df-frec 6405

This theorem is referenced by:  freccl  6417