Theorem freccllem 6400

Description: Lemma for freccl 6401. 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 6389	. . . 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 4902	. . . 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 2201	. . 3 ⊢ frec(𝐹, 𝐴) = (𝐺 ↾ ω)
5	4	fveq1i 5515	. 2 ⊢ (frec(𝐹, 𝐴)‘𝐵) = ((𝐺 ↾ ω)‘𝐵)
6		freccl.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ω)
7		fvres 5538	. . . 4 ⊢ (𝐵 ∈ ω → ((𝐺 ↾ ω)‘𝐵) = (𝐺‘𝐵))
8	6, 7	syl 14	. . 3 ⊢ (𝜑 → ((𝐺 ↾ ω)‘𝐵) = (𝐺‘𝐵))
9		funmpt 5253	. . . . 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 4605	. . . . 5 ⊢ Ord ω
12	11	a1i 9	. . . 4 ⊢ (𝜑 → Ord ω)
13		vex 2740	. . . . . 6 ⊢ 𝑓 ∈ V
14		simp2 998	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑦 ∈ ω)
15		simp3 999	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑓:𝑦⟶𝑆)
16		freccl.cl	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝑆) → (𝐹‘𝑧) ∈ 𝑆)
17	16	ralrimiva 2550	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆)
18	17	3ad2ant1 1018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆)
19		freccl.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑆)
20	19	3ad2ant1 1018	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝐴 ∈ 𝑆)
21	14, 15, 18, 20	frecabcl 6397	. . . . . 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 4826	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
23	22	eqeq1d 2186	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
24		fveq1 5513	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑚) = (𝑓‘𝑚))
25	24	fveq2d 5518	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(𝑓‘𝑚)))
26	25	eleq2d 2247	. . . . . . . . . . 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 2478	. . . . . . . . 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 2186	. . . . . . . . . 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 793	. . . . . . . 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 2295	. . . . . . 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 2177	. . . . . . 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 5591	. . . . . 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 2254	. . . 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 4612	. . . . . . 7 ⊢ Lim ω
38		limuni 4395	. . . . . . 7 ⊢ (Lim ω → ω = ∪ ω)
39	37, 38	ax-mp 5	. . . . . 6 ⊢ ω = ∪ ω
40	39	eleq2i 2244	. . . . 5 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∪ ω)
41		peano2 4593	. . . . . 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 2270	. . . 4 ⊢ (𝜑 → 𝐵 ∈ ∪ ω)
45	2, 10, 12, 36, 43, 44	tfrcl 6362	. . 3 ⊢ (𝜑 → (𝐺‘𝐵) ∈ 𝑆)
46	8, 45	eqeltrd 2254	. 2 ⊢ (𝜑 → ((𝐺 ↾ ω)‘𝐵) ∈ 𝑆)
47	5, 46	eqeltrid 2264	1 ⊢ (𝜑 → (frec(𝐹, 𝐴)‘𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  Vcvv 2737  ∅c0 3422  ∪ cuni 3809   ↦ cmpt 4063  Ord word 4361  Lim wlim 4363  suc csuc 4364  ωcom 4588  dom cdm 4625   ↾ cres 4627  Fun wfun 5209  ⟶wf 5211  ‘cfv 5215  recscrecs 6302  freccfrec 6388

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 4117  ax-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-iinf 4586

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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-tr 4101  df-id 4292  df-iord 4365  df-on 4367  df-ilim 4368  df-suc 4370  df-iom 4589  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-recs 6303  df-frec 6389

This theorem is referenced by:  freccl  6401