Theorem freccllem 6559

Description: Lemma for freccl 6560. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 27-Mar-2022.)

Hypotheses

Ref	Expression
freccl.a	⊢ (𝜑 → 𝐴 ∈ 𝑆)
freccl.cl	⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)
freccl.b	⊢ (𝜑 → 𝐵 ∈ ω)
freccllem.g	⊢ 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))

Assertion

Ref	Expression
freccllem	⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

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

Allowed substitution hints:   𝜑(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 6548	. . . 4 ⊢ frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
2		freccllem.g	. . . . 5 ⊢ 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
3	2	reseq1i 5004	. . . 4 ⊢ (𝐺 ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
4	1, 3	eqtr4i 2253	. . 3 ⊢ frec(𝐹, 𝐴) = (𝐺 ↾ ω)
5	4	fveq1i 5633	. 2 ⊢ (frec(𝐹, 𝐴)‘𝐵) = ((𝐺 ↾ ω)‘𝐵)
6		freccl.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ω)
7		fvres 5656	. . . 4 ⊢ (𝐵 ∈ ω → ((𝐺 ↾ ω)‘𝐵) = (𝐺‘𝐵))
8	6, 7	syl 14	. . 3 ⊢ (𝜑 → ((𝐺 ↾ ω)‘𝐵) = (𝐺‘𝐵))
9		funmpt 5359	. . . . 5 ⊢ Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
10	9	a1i 9	. . . 4 ⊢ (𝜑 → Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
11		ordom 4700	. . . . 5 ⊢ Ord ω
12	11	a1i 9	. . . 4 ⊢ (𝜑 → Ord ω)
13		vex 2802	. . . . . 6 ⊢ 𝑓 ∈ V
14		simp2 1022	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑦 ∈ ω)
15		simp3 1023	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑓:𝑦⟶𝑆)
16		freccl.cl	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)
17	16	ralrimiva 2603	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆)
18	17	3ad2ant1 1042	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆)
19		freccl.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
20	19	3ad2ant1 1042	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝐴 ∈ 𝑆)
21	14, 15, 18, 20	frecabcl 6556	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ 𝑆)
22		dmeq 4926	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
23	22	eqeq1d 2238	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
24		fveq1 5631	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑚) = (𝑓‘𝑚))
25	24	fveq2d 5636	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(𝑓‘𝑚)))
26	25	eleq2d 2299	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (𝑥 ∈ (𝐹‘(𝑔‘𝑚)) ↔ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))))
27	23, 26	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → ((dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚)))))
28	27	rexbidv 2531	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚)))))
29	22	eqeq1d 2238	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
30	29	anbi1d 465	. . . . . . . . 9 ⊢ (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴)))
31	28, 30	orbi12d 798	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → ((∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴)) ↔ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))))
32	31	abbidv 2347	. . . . . . 7 ⊢ (𝑔 = 𝑓 → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))} = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))})
33		eqid 2229	. . . . . . 7 ⊢ (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
34	32, 33	fvmptg 5715	. . . . . 6 ⊢ ((𝑓 ∈ V ∧ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ 𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘𝑓) = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))})
35	13, 21, 34	sylancr 414	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘𝑓) = {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))})
36	35, 21	eqeltrd 2306	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘𝑓) ∈ 𝑆)
37		limom 4707	. . . . . . 7 ⊢ Lim ω
38		limuni 4488	. . . . . . 7 ⊢ (Lim ω → ω = ∪ ω)
39	37, 38	ax-mp 5	. . . . . 6 ⊢ ω = ∪ ω
40	39	eleq2i 2296	. . . . 5 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∪ ω)
41		peano2 4688	. . . . . 6 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
42	41	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ω) → suc 𝑦 ∈ ω)
43	40, 42	sylan2br 288	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ ∪ ω) → suc 𝑦 ∈ ω)
44	6, 39	eleqtrdi 2322	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ∪ ω)
45	2, 10, 12, 36, 43, 44	tfrcl 6521	. . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ 𝑆)
46	8, 45	eqeltrd 2306	. 2 ⊢ (𝜑 → ((𝐺 ↾ ω)‘𝐵) ∈ 𝑆)
47	5, 46	eqeltrid 2316	1 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 713   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  {cab 2215  ∀wral 2508  ∃wrex 2509  Vcvv 2799  ∅c0 3491  ∪ cuni 3888   ↦ cmpt 4145  Ord word 4454  Lim wlim 4456  suc csuc 4457  ωcom 4683  dom cdm 4720   ↾ cres 4722  Fun wfun 5315  ⟶wf 5317  ‘cfv 5321  recscrecs 6461  freccfrec 6547

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681

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 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-recs 6462  df-frec 6548

This theorem is referenced by:  freccl  6560