Theorem freccllem 6546

Description: Lemma for freccl 6547. 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 6535	. . . 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 5000	. . . 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 2253	. . 3 |- frec ( F , A ) = ( G |` om )
5	4	fveq1i 5627	. 2 |- (frec ( F , A ) ` B ) = ( ( G |` om ) ` B )
6		freccl.b	. . . 4 |- ( ph -> B e. om )
7		fvres 5650	. . . 4 |- ( B e. om -> ( ( G |` om ) ` B ) = ( G ` B ) )
8	6, 7	syl 14	. . 3 |- ( ph -> ( ( G |` om ) ` B ) = ( G ` B ) )
9		funmpt 5355	. . . . 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 4698	. . . . 5 |- Ord om
12	11	a1i 9	. . . 4 |- ( ph -> Ord om )
13		vex 2802	. . . . . 6 |- f e. _V
14		simp2 1022	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> y e. om )
15		simp3 1023	. . . . . . 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 2603	. . . . . . . 8 |- ( ph -> A. z e. S ( F ` z ) e. S )
18	17	3ad2ant1 1042	. . . . . . 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 1042	. . . . . . 7 |- ( ( ph /\ y e. om /\ f : y --> S ) -> A e. S )
21	14, 15, 18, 20	frecabcl 6543	. . . . . 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 4922	. . . . . . . . . . . 12 |- ( g = f -> dom g = dom f )
23	22	eqeq1d 2238	. . . . . . . . . . 11 |- ( g = f -> ( dom g = suc m <-> dom f = suc m ) )
24		fveq1 5625	. . . . . . . . . . . . 13 |- ( g = f -> ( g ` m ) = ( f ` m ) )
25	24	fveq2d 5630	. . . . . . . . . . . 12 |- ( g = f -> ( F ` ( g ` m ) ) = ( F ` ( f ` m ) ) )
26	25	eleq2d 2299	. . . . . . . . . . 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 2531	. . . . . . . . 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 2238	. . . . . . . . . 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 798	. . . . . . . 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 2347	. . . . . . 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 2229	. . . . . . 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 5709	. . . . . 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 2306	. . . 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 4705	. . . . . . 7 |- Lim om
38		limuni 4486	. . . . . . 7 |- ( Lim om -> om = U. om )
39	37, 38	ax-mp 5	. . . . . 6 |- om = U. om
40	39	eleq2i 2296	. . . . 5 |- ( y e. om <-> y e. U. om )
41		peano2 4686	. . . . . 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 2322	. . . 4 |- ( ph -> B e. U. om )
45	2, 10, 12, 36, 43, 44	tfrcl 6508	. . 3 |- ( ph -> ( G ` B ) e. S )
46	8, 45	eqeltrd 2306	. 2 |- ( ph -> ( ( G |` om ) ` B ) e. S )
47	5, 46	eqeltrid 2316	1 |- ( ph -> (frec ( F , A ) ` B ) e. S )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   {cab 2215   A.wral 2508   E.wrex 2509   _Vcvv 2799   (/)c0 3491   U.cuni 3887   |-> cmpt 4144   Ord word 4452   Lim wlim 4454   suc csuc 4455   omcom 4681   dom cdm 4718   |` cres 4720   Fun wfun 5311   -->wf 5313   ` cfv 5317  recscrecs 6448  freccfrec 6534

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-recs 6449  df-frec 6535

This theorem is referenced by:  freccl  6547