Theorem frecfcllem 6372

Description: Lemma for frecfcl 6373. 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 5226	. . . . . . 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 4584	. . . . . . 7 ⊢ Ord ω
5	4	a1i 9	. . . . . 6 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → Ord ω)
6		vex 2729	. . . . . . . 8 ⊢ 𝑓 ∈ V
7		simp2 988	. . . . . . . . 9 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑦 ∈ ω)
8		simp3 989	. . . . . . . . 9 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝑓:𝑦⟶𝑆)
9		simp1ll 1050	. . . . . . . . . 10 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆)
10		fveq2 5486	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
11	10	eleq1d 2235	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) ∈ 𝑆 ↔ (𝐹‘𝑤) ∈ 𝑆))
12	11	cbvralv 2692	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ↔ ∀𝑤 ∈ 𝑆 (𝐹‘𝑤) ∈ 𝑆)
13	9, 12	sylib 121	. . . . . . . . 9 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → ∀𝑤 ∈ 𝑆 (𝐹‘𝑤) ∈ 𝑆)
14		simp1lr 1051	. . . . . . . . 9 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ω ∧ 𝑓:𝑦⟶𝑆) → 𝐴 ∈ 𝑆)
15	7, 8, 13, 14	frecabcl 6367	. . . . . . . 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 4804	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
17	16	eqeq1d 2174	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (dom 𝑔 = suc 𝑚 ↔ dom 𝑓 = suc 𝑚))
18		fveq1 5485	. . . . . . . . . . . . . . 15 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑚) = (𝑓‘𝑚))
19	18	fveq2d 5490	. . . . . . . . . . . . . 14 ⊢ (𝑔 = 𝑓 → (𝐹‘(𝑔‘𝑚)) = (𝐹‘(𝑓‘𝑚)))
20	19	eleq2d 2236	. . . . . . . . . . . . 13 ⊢ (𝑔 = 𝑓 → (𝑥 ∈ (𝐹‘(𝑔‘𝑚)) ↔ 𝑥 ∈ (𝐹‘(𝑓‘𝑚))))
21	17, 20	anbi12d 465	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → ((dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ↔ (dom 𝑓 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑓‘𝑚)))))
22	21	rexbidv 2467	. . . . . . . . . . 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 2174	. . . . . . . . . . . 12 ⊢ (𝑔 = 𝑓 → (dom 𝑔 = ∅ ↔ dom 𝑓 = ∅))
24	23	anbi1d 461	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → ((dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴) ↔ (dom 𝑓 = ∅ ∧ 𝑥 ∈ 𝐴)))
25	22, 24	orbi12d 783	. . . . . . . . . 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 2284	. . . . . . . . 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 2165	. . . . . . . . 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 5562	. . . . . . . 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 411	. . . . . . 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 2243	. . . . . 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 4591	. . . . . . . . . 10 ⊢ Lim ω
32		limuni 4374	. . . . . . . . . 10 ⊢ (Lim ω → ω = ∪ ω)
33	31, 32	ax-mp 5	. . . . . . . . 9 ⊢ ω = ∪ ω
34	33	eleq2i 2233	. . . . . . . 8 ⊢ (𝑦 ∈ ω ↔ 𝑦 ∈ ∪ ω)
35		peano2 4572	. . . . . . . 8 ⊢ (𝑦 ∈ ω → suc 𝑦 ∈ ω)
36	34, 35	sylbir 134	. . . . . . 7 ⊢ (𝑦 ∈ ∪ ω → suc 𝑦 ∈ ω)
37	36	adantl 275	. . . . . 6 ⊢ ((((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) ∧ 𝑦 ∈ ∪ ω) → suc 𝑦 ∈ ω)
38	33	eleq2i 2233	. . . . . . . 8 ⊢ (𝑘 ∈ ω ↔ 𝑘 ∈ ∪ ω)
39	38	biimpi 119	. . . . . . 7 ⊢ (𝑘 ∈ ω → 𝑘 ∈ ∪ ω)
40	39	adantl 275	. . . . . 6 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → 𝑘 ∈ ∪ ω)
41	1, 3, 5, 30, 37, 40	tfrcldm 6331	. . . . 5 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → 𝑘 ∈ dom 𝐺)
42	1, 3, 5, 30, 37, 40	tfrcl 6332	. . . . 5 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → (𝐺‘𝑘) ∈ 𝑆)
43	41, 42	jca 304	. . . 4 ⊢ (((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ∧ 𝑘 ∈ ω) → (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑆))
44	43	ralrimiva 2539	. . 3 ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ∀𝑘 ∈ ω (𝑘 ∈ dom 𝐺 ∧ (𝐺‘𝑘) ∈ 𝑆))
45		tfrfun 6288	. . . . 5 ⊢ Fun recs((𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}))
46	1	funeqi 5209	. . . . 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 145	. . . 4 ⊢ Fun 𝐺
48		ffvresb 5648	. . . 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 133	. 2 ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐺 ↾ ω):ω⟶𝑆)
51		df-frec 6359	. . . 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 4880	. . . 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 2189	. . 3 ⊢ frec(𝐹, 𝐴) = (𝐺 ↾ ω)
54	53	feq1i 5330	. 2 ⊢ (frec(𝐹, 𝐴):ω⟶𝑆 ↔ (𝐺 ↾ ω):ω⟶𝑆)
55	50, 54	sylibr 133	1 ⊢ ((∀𝑧 ∈ 𝑆 (𝐹‘𝑧) ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → frec(𝐹, 𝐴):ω⟶𝑆)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∃wrex 2445  Vcvv 2726  ∅c0 3409  ∪ cuni 3789   ↦ cmpt 4043  Ord word 4340  Lim wlim 4342  suc csuc 4343  ωcom 4567  dom cdm 4604   ↾ cres 4606  Fun wfun 5182  ⟶wf 5184  ‘cfv 5188  recscrecs 6272  freccfrec 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-recs 6273  df-frec 6359

This theorem is referenced by:  frecfcl  6373