Theorem freccllem 6632

Description: Lemma for freccl 6633. 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 6621	. . . 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 5033	. . . 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 2256	. . 3 ⊢ frec(𝐹, 𝐴) = (𝐺 ↾ ω)
5	4	fveq1i 5670	. 2 ⊢ (frec(𝐹, 𝐴)‘𝐵) = ((𝐺 ↾ ω)‘𝐵)
6		freccl.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ω)
7		fvres 5693	. . . 4 ⊢ (𝐵 ∈ ω → ((𝐺 ↾ ω)‘𝐵) = (𝐺‘𝐵))
8	6, 7	syl 14	. . 3 ⊢ (𝜑 → ((𝐺 ↾ ω)‘𝐵) = (𝐺‘𝐵))
9		funmpt 5389	. . . . 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 4728	. . . . 5 ⊢ Ord ω
12	11	a1i 9	. . . 4 ⊢ (𝜑 → Ord ω)
13		vex 2815	. . . . . 6 ⊢ 𝑓 ∈ V
14		simp2 1025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑦 ∈ ω)
15		simp3 1026	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑓:𝑦⟶𝑆)
16		freccl.cl	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)
17	16	ralrimiva 2615	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆)
18	17	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆)
19		freccl.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
20	19	3ad2ant1 1045	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝐴 ∈ 𝑆)
21	14, 15, 18, 20	frecabcl 6629	. . . . . 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 4955	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
23	22	eqeq1d 2241	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
24		fveq1 5668	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑚) = (𝑓‘𝑚))
25	24	fveq2d 5673	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(𝑓‘𝑚)))
26	25	eleq2d 2302	. . . . . . . . . . 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 2543	. . . . . . . . 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 2241	. . . . . . . . . 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 801	. . . . . . . 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 2352	. . . . . . 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 2232	. . . . . . 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 5752	. . . . . 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 2309	. . . 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 4735	. . . . . . 7 ⊢ Lim ω
38		limuni 4516	. . . . . . 7 ⊢ (Lim ω → ω = ∪ ω)
39	37, 38	ax-mp 5	. . . . . 6 ⊢ ω = ∪ ω
40	39	eleq2i 2299	. . . . 5 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∪ ω)
41		peano2 4716	. . . . . 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 2325	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ∪ ω)
45	2, 10, 12, 36, 43, 44	tfrcl 6594	. . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ 𝑆)
46	8, 45	eqeltrd 2309	. 2 ⊢ (𝜑 → ((𝐺 ↾ ω)‘𝐵) ∈ 𝑆)
47	5, 46	eqeltrid 2319	1 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 716   ∧ w3a 1005   = wceq 1398   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  Vcvv 2812  ∅c0 3507  ∪ cuni 3913   ↦ cmpt 4170  Ord word 4482  Lim wlim 4484  suc csuc 4485  ωcom 4711  dom cdm 4748   ↾ cres 4750  Fun wfun 5345  ⟶wf 5347  ‘cfv 5351  recscrecs 6534  freccfrec 6620

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-recs 6535  df-frec 6621

This theorem is referenced by:  freccl  6633