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Theorem ctiunctlemudc 12370

Description: Lemma for ctiunct 12373. (Contributed by Jim Kingdon, 28-Oct-2023.)

**Hypotheses**
Ref	Expression
ctiunct.som	⊢ (𝜑 → 𝑆 ⊆ ω)
ctiunct.sdc	⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
ctiunct.f	⊢ (𝜑 → 𝐹:𝑆–onto→𝐴)
ctiunct.tom	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑇 ⊆ ω)
ctiunct.tdc	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
ctiunct.g	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐺:𝑇–onto→𝐵)
ctiunct.j	⊢ (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))
ctiunct.u	⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}

**Assertion**
Ref	Expression
ctiunctlemudc	⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑈)

Distinct variable groups: 𝑥,𝐴 𝑛,𝐹,𝑥 𝑧,𝐹,𝑥 𝑛,𝐽,𝑥 𝑧,𝐽 𝑆,𝑛 𝑧,𝑆 𝑇,𝑛 𝑧,𝑇 𝑈,𝑛 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑧,𝑛) 𝐴(𝑧,𝑛) 𝐵(𝑥,𝑧,𝑛) 𝑆(𝑥) 𝑇(𝑥) 𝑈(𝑥,𝑧) 𝐺(𝑥,𝑧,𝑛)

Proof of Theorem ctiunctlemudc Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2229	. . . . . . . . 9 ⊢ (𝑛 = (1^st ‘(𝐽‘𝑚)) → (𝑛 ∈ 𝑆 ↔ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆))
2	1	dcbid 828	. . . . . . . 8 ⊢ (𝑛 = (1^st ‘(𝐽‘𝑚)) → (DECID 𝑛 ∈ 𝑆 ↔ DECID (1^st ‘(𝐽‘𝑚)) ∈ 𝑆))
3		ctiunct.sdc	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
4	3	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑆)
5		ctiunct.j	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐽:ω–1-1-onto→(ω × ω))
6	5	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝐽:ω–1-1-onto→(ω × ω))
7		f1of 5432	. . . . . . . . . . 11 ⊢ (𝐽:ω–1-1-onto→(ω × ω) → 𝐽:ω⟶(ω × ω))
8	6, 7	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝐽:ω⟶(ω × ω))
9		simpr 109	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → 𝑚 ∈ ω)
10	8, 9	ffvelrnd 5621	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (𝐽‘𝑚) ∈ (ω × ω))
11		xp1st 6133	. . . . . . . . 9 ⊢ ((𝐽‘𝑚) ∈ (ω × ω) → (1^st ‘(𝐽‘𝑚)) ∈ ω)
12	10, 11	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (1^st ‘(𝐽‘𝑚)) ∈ ω)
13	2, 4, 12	rspcdva 2835	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → DECID (1^st ‘(𝐽‘𝑚)) ∈ 𝑆)
14	13	adantr 274	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → DECID (1^st ‘(𝐽‘𝑚)) ∈ 𝑆)
15		eleq1 2229	. . . . . . . 8 ⊢ (𝑛 = (2^nd ‘(𝐽‘𝑚)) → (𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇 ↔ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
16	15	dcbid 828	. . . . . . 7 ⊢ (𝑛 = (2^nd ‘(𝐽‘𝑚)) → (DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇 ↔ DECID (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
17		ctiunct.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹:𝑆–onto→𝐴)
18		fof 5410	. . . . . . . . . . 11 ⊢ (𝐹:𝑆–onto→𝐴 → 𝐹:𝑆⟶𝐴)
19	17, 18	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑆⟶𝐴)
20	19	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → 𝐹:𝑆⟶𝐴)
21		simpr 109	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → (1^st ‘(𝐽‘𝑚)) ∈ 𝑆)
22	20, 21	ffvelrnd 5621	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → (𝐹‘(1^st ‘(𝐽‘𝑚))) ∈ 𝐴)
23		ctiunct.tdc	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
24	23	ralrimiva 2539	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
25	24	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → ∀𝑥 ∈ 𝐴 ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇)
26		nfcv 2308	. . . . . . . . . 10 ⊢ Ⅎ𝑥ω
27		nfcsb1v 3078	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇
28	27	nfcri 2302	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇
29	28	nfdc 1647	. . . . . . . . . 10 ⊢ Ⅎ𝑥DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇
30	26, 29	nfralya 2506	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑛 ∈ ω DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇
31		csbeq1a 3054	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘(1^st ‘(𝐽‘𝑚))) → 𝑇 = ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)
32	31	eleq2d 2236	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘(1^st ‘(𝐽‘𝑚))) → (𝑛 ∈ 𝑇 ↔ 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
33	32	dcbid 828	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘(1^st ‘(𝐽‘𝑚))) → (DECID 𝑛 ∈ 𝑇 ↔ DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
34	33	ralbidv 2466	. . . . . . . . 9 ⊢ (𝑥 = (𝐹‘(1^st ‘(𝐽‘𝑚))) → (∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
35	30, 34	rspc 2824	. . . . . . . 8 ⊢ ((𝐹‘(1^st ‘(𝐽‘𝑚))) ∈ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑇 → ∀𝑛 ∈ ω DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
36	22, 25, 35	sylc 62	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → ∀𝑛 ∈ ω DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)
37	10	adantr 274	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → (𝐽‘𝑚) ∈ (ω × ω))
38		xp2nd 6134	. . . . . . . 8 ⊢ ((𝐽‘𝑚) ∈ (ω × ω) → (2^nd ‘(𝐽‘𝑚)) ∈ ω)
39	37, 38	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → (2^nd ‘(𝐽‘𝑚)) ∈ ω)
40	16, 36, 39	rspcdva 2835	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → DECID (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)
41		dcan2 924	. . . . . 6 ⊢ (DECID (1^st ‘(𝐽‘𝑚)) ∈ 𝑆 → (DECID (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇 → DECID ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)))
42	14, 40, 41	sylc 62	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → DECID ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
43		simpr 109	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ ¬ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → ¬ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆)
44	43	intnanrd 922	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ ¬ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → ¬ ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
45	44	olcd 724	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ ¬ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → (((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇) ∨ ¬ ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)))
46		df-dc 825	. . . . . 6 ⊢ (DECID ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇) ↔ (((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇) ∨ ¬ ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)))
47	45, 46	sylibr 133	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ ¬ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → DECID ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
48		exmiddc 826	. . . . . 6 ⊢ (DECID (1^st ‘(𝐽‘𝑚)) ∈ 𝑆 → ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∨ ¬ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆))
49	13, 48	syl 14	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∨ ¬ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆))
50	42, 47, 49	mpjaodan 788	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → DECID ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
51		2fveq3 5491	. . . . . . . . 9 ⊢ (𝑧 = 𝑚 → (1^st ‘(𝐽‘𝑧)) = (1^st ‘(𝐽‘𝑚)))
52	51	eleq1d 2235	. . . . . . . 8 ⊢ (𝑧 = 𝑚 → ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ↔ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆))
53		2fveq3 5491	. . . . . . . . 9 ⊢ (𝑧 = 𝑚 → (2^nd ‘(𝐽‘𝑧)) = (2^nd ‘(𝐽‘𝑚)))
54	51	fveq2d 5490	. . . . . . . . . 10 ⊢ (𝑧 = 𝑚 → (𝐹‘(1^st ‘(𝐽‘𝑧))) = (𝐹‘(1^st ‘(𝐽‘𝑚))))
55	54	csbeq1d 3052	. . . . . . . . 9 ⊢ (𝑧 = 𝑚 → ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 = ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)
56	53, 55	eleq12d 2237	. . . . . . . 8 ⊢ (𝑧 = 𝑚 → ((2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 ↔ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
57	52, 56	anbi12d 465	. . . . . . 7 ⊢ (𝑧 = 𝑚 → (((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇) ↔ ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)))
58		ctiunct.u	. . . . . . 7 ⊢ 𝑈 = {𝑧 ∈ ω ∣ ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑧)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇)}
59	57, 58	elrab2 2885	. . . . . 6 ⊢ (𝑚 ∈ 𝑈 ↔ (𝑚 ∈ ω ∧ ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)))
60		ibar 299	. . . . . . 7 ⊢ (𝑚 ∈ ω → (((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇) ↔ (𝑚 ∈ ω ∧ ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))))
61	60	adantl 275	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇) ↔ (𝑚 ∈ ω ∧ ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))))
62	59, 61	bitr4id 198	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (𝑚 ∈ 𝑈 ↔ ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)))
63	62	dcbid 828	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (DECID 𝑚 ∈ 𝑈 ↔ DECID ((1^st ‘(𝐽‘𝑚)) ∈ 𝑆 ∧ (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)))
64	50, 63	mpbird 166	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → DECID 𝑚 ∈ 𝑈)
65	64	ralrimiva 2539	. 2 ⊢ (𝜑 → ∀𝑚 ∈ ω DECID 𝑚 ∈ 𝑈)
66		eleq1 2229	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ 𝑈 ↔ 𝑛 ∈ 𝑈))
67	66	dcbid 828	. . 3 ⊢ (𝑚 = 𝑛 → (DECID 𝑚 ∈ 𝑈 ↔ DECID 𝑛 ∈ 𝑈))
68	67	cbvralv 2692	. 2 ⊢ (∀𝑚 ∈ ω DECID 𝑚 ∈ 𝑈 ↔ ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑈)
69	65, 68	sylib 121	1 ⊢ (𝜑 → ∀𝑛 ∈ ω DECID 𝑛 ∈ 𝑈)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 824 = wceq 1343 ∈ wcel 2136 ∀wral 2444 {crab 2448 ⦋csb 3045 ⊆ wss 3116 ωcom 4567 × cxp 4602 ⟶wf 5184 –onto→wfo 5186 –1-1-onto→wf1o 5187 ‘cfv 5188 1^st c1st 6106 2^nd c2nd 6107

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-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411

This theorem depends on definitions: df-bi 116 df-dc 825 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-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 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-1st 6108 df-2nd 6109

This theorem is referenced by: ctiunct 12373

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