Theorem frecfcllem 6407

Description: Lemma for frecfcl 6408. Just giving a name to a common expression to simplify the proof. (Contributed by Jim Kingdon, 30-Mar-2022.)

Hypothesis

Ref	Expression
frecfcllem.g	⊢ 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))

Assertion

Ref	Expression
frecfcllem	⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)

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

Allowed substitution hints:   𝐴(𝑧)   𝑆(𝑔)   𝐺(𝑥,𝑧,𝑔,𝑚)

Proof of Theorem frecfcllem

Dummy variables 𝑓 𝑦 𝑘 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frecfcllem.g	. . . . . 6 ⊢ 𝐺 = recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
2		funmpt 5256	. . . . . . 7 ⊢ Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})
3	2	a1i 9	. . . . . 6 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → Fun (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
4		ordom 4608	. . . . . . 7 ⊢ Ord ω
5	4	a1i 9	. . . . . 6 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → Ord ω)
6		vex 2742	. . . . . . . 8 ⊢ 𝑓 ∈ V
7		simp2 998	. . . . . . . . 9 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑦 ∈ ω)
8		simp3 999	. . . . . . . . 9 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑓:𝑦⟶𝑆)
9		simp1ll 1060	. . . . . . . . . 10 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆)
10		fveq2 5517	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
11	10	eleq1d 2246	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ∈ 𝑆 ↔ (𝐹‘𝑤) ∈ 𝑆))
12	11	cbvralv 2705	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ↔ ∀𝑤 ∈ 𝑆 (𝐹‘𝑤) ∈ 𝑆)
13	9, 12	sylib 122	. . . . . . . . 9 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ∀𝑤 ∈ 𝑆 (𝐹‘𝑤) ∈ 𝑆)
14		simp1lr 1061	. . . . . . . . 9 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝐴 ∈ 𝑆)
15	7, 8, 13, 14	frecabcl 6402	. . . . . . . 8 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))) ∨ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴))} ∈ 𝑆)
16		dmeq 4829	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
17	16	eqeq1d 2186	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
18		fveq1 5516	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑚) = (𝑓‘𝑚))
19	18	fveq2d 5521	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(𝑓‘𝑚)))
20	19	eleq2d 2247	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝑥 ∈ (𝐹‘(𝑔‘𝑚)) ↔ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))))
21	17, 20	anbi12d 473	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚)))))
22	21	rexbidv 2478	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ ∃𝑚 ∈ ω (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚)))))
23	16	eqeq1d 2186	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
24	23	anbi1d 465	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴)))
25	22, 24	orbi12d 793	. . . . . . . . . 10 ⊢ (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))))
26	25	abbidv 2295	. . . . . . . . 9 ⊢ (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))})
27		eqid 2177	. . . . . . . . 9 ⊢ (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))})
28	26, 27	fvmptg 5594	. . . . . . . 8 ⊢ ((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))})
29	6, 15, 28	sylancr 414	. . . . . . 7 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 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))})
30	29, 15	eqeltrd 2254	. . . . . 6 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})‘𝑓) ∈ 𝑆)
31		limom 4615	. . . . . . . . . 10 ⊢ Lim ω
32		limuni 4398	. . . . . . . . . 10 ⊢ (Lim ω → ω = ∪ ω)
33	31, 32	ax-mp 5	. . . . . . . . 9 ⊢ ω = ∪ ω
34	33	eleq2i 2244	. . . . . . . 8 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∪ ω)
35		peano2 4596	. . . . . . . 8 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
36	34, 35	sylbir 135	. . . . . . 7 ⊢ (𝑦 ∈ ∪ ω → suc 𝑦 ∈ ω)
37	36	adantl 277	. . . . . 6 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ∪ ω) → suc 𝑦 ∈ ω)
38	33	eleq2i 2244	. . . . . . . 8 ⊢ (𝑘 ∈ ω ↔ 𝑘 ∈ ∪ ω)
39	38	biimpi 120	. . . . . . 7 ⊢ (𝑘 ∈ ω → 𝑘 ∈ ∪ ω)
40	39	adantl 277	. . . . . 6 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → 𝑘 ∈ ∪ ω)
41	1, 3, 5, 30, 37, 40	tfrcldm 6366	. . . . 5 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → 𝑘 ∈ dom 𝐺)
42	1, 3, 5, 30, 37, 40	tfrcl 6367	. . . . 5 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → (𝐺‘𝑘) ∈ 𝑆)
43	41, 42	jca 306	. . . 4 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑆))
44	43	ralrimiva 2550	. . 3 ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑆))
45		tfrfun 6323	. . . . 5 ⊢ Fun recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
46	1	funeqi 5239	. . . . 5 ⊢ (Fun 𝐺 ↔ Fun recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})))
47	45, 46	mpbir 146	. . . 4 ⊢ Fun 𝐺
48		ffvresb 5681	. . . 4 ⊢ (Fun 𝐺 → ((𝐺 ↾ ω):ω⟶𝑆 ↔ ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑆)))
49	47, 48	ax-mp 5	. . 3 ⊢ ((𝐺 ↾ ω):ω⟶𝑆 ↔ ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑆))
50	44, 49	sylibr 134	. 2 ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐺 ↾ ω):ω⟶𝑆)
51		df-frec 6394	. . . 4 ⊢ frec(𝐹, 𝐴) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
52	1	reseq1i 4905	. . . 4 ⊢ (𝐺 ↾ ω) = (recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))})) ↾ ω)
53	51, 52	eqtr4i 2201	. . 3 ⊢ frec(𝐹, 𝐴) = (𝐺 ↾ ω)
54	53	feq1i 5360	. 2 ⊢ (frec(𝐹, 𝐴):ω⟶𝑆 ↔ (𝐺 ↾ ω):ω⟶𝑆)
55	50, 54	sylibr 134	1 ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  Vcvv 2739  ∅c0 3424  ∪ cuni 3811   ↦ cmpt 4066  Ord word 4364  Lim wlim 4366  suc csuc 4367  ωcom 4591  dom cdm 4628   ↾ cres 4630  Fun wfun 5212  ⟶wf 5214  ‘cfv 5218  recscrecs 6307  freccfrec 6393

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-recs 6308  df-frec 6394

This theorem is referenced by:  frecfcl  6408