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Theorem ctiunctlemudc 11986

Description: Lemma for ctiunct 11989. (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 2203	. . . . . . . . 9 ⊢ (𝑛 = (1^st ‘(𝐽‘𝑚)) → (𝑛 ∈ 𝑆 ↔ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆))
2	1	dcbid 824	. . . . . . . 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 5375	. . . . . . . . . . 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 5564	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ ω) → (𝐽‘𝑚) ∈ (ω × ω))
11		xp1st 6071	. . . . . . . . 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 2798	. . . . . . 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 2203	. . . . . . . 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 824	. . . . . . 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 5353	. . . . . . . . . . 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 5564	. . . . . . . 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 2508	. . . . . . . . 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 2282	. . . . . . . . . 10 ⊢ Ⅎ𝑥ω
27		nfcsb1v 3040	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇
28	27	nfcri 2276	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇
29	28	nfdc 1638	. . . . . . . . . 10 ⊢ Ⅎ𝑥DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇
30	26, 29	nfralya 2476	. . . . . . . . 9 ⊢ Ⅎ𝑥∀𝑛 ∈ ω DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇
31		csbeq1a 3016	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝐹‘(1^st ‘(𝐽‘𝑚))) → 𝑇 = ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)
32	31	eleq2d 2210	. . . . . . . . . . 11 ⊢ (𝑥 = (𝐹‘(1^st ‘(𝐽‘𝑚))) → (𝑛 ∈ 𝑇 ↔ 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
33	32	dcbid 824	. . . . . . . . . 10 ⊢ (𝑥 = (𝐹‘(1^st ‘(𝐽‘𝑚))) → (DECID 𝑛 ∈ 𝑇 ↔ DECID 𝑛 ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇))
34	33	ralbidv 2438	. . . . . . . . 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 2787	. . . . . . . 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 6072	. . . . . . . 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 2798	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ω) ∧ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆) → DECID (2^nd ‘(𝐽‘𝑚)) ∈ ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)
41		dcan 919	. . . . . 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 918	. . . . . . 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 821	. . . . . 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 822	. . . . . 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 5434	. . . . . . . . 9 ⊢ (𝑧 = 𝑚 → (1^st ‘(𝐽‘𝑧)) = (1^st ‘(𝐽‘𝑚)))
52	51	eleq1d 2209	. . . . . . . 8 ⊢ (𝑧 = 𝑚 → ((1^st ‘(𝐽‘𝑧)) ∈ 𝑆 ↔ (1^st ‘(𝐽‘𝑚)) ∈ 𝑆))
53		2fveq3 5434	. . . . . . . . 9 ⊢ (𝑧 = 𝑚 → (2^nd ‘(𝐽‘𝑧)) = (2^nd ‘(𝐽‘𝑚)))
54	51	fveq2d 5433	. . . . . . . . . 10 ⊢ (𝑧 = 𝑚 → (𝐹‘(1^st ‘(𝐽‘𝑧))) = (𝐹‘(1^st ‘(𝐽‘𝑚))))
55	54	csbeq1d 3014	. . . . . . . . 9 ⊢ (𝑧 = 𝑚 → ⦋(𝐹‘(1^st ‘(𝐽‘𝑧))) / 𝑥⦌𝑇 = ⦋(𝐹‘(1^st ‘(𝐽‘𝑚))) / 𝑥⦌𝑇)
56	53, 55	eleq12d 2211	. . . . . . . 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 2847	. . . . . 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 824	. . . 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 2508	. 2 ⊢ (𝜑 → ∀𝑚 ∈ ω DECID 𝑚 ∈ 𝑈)
66		eleq1 2203	. . . 4 ⊢ (𝑚 = 𝑛 → (𝑚 ∈ 𝑈 ↔ 𝑛 ∈ 𝑈))
67	66	dcbid 824	. . 3 ⊢ (𝑚 = 𝑛 → (DECID 𝑚 ∈ 𝑈 ↔ DECID 𝑛 ∈ 𝑈))
68	67	cbvralv 2657	. 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 820 = wceq 1332 ∈ wcel 1481 ∀wral 2417 {crab 2421 ⦋csb 3007 ⊆ wss 3076 ωcom 4512 × cxp 4545 ⟶wf 5127 –onto→wfo 5129 –1-1-onto→wf1o 5130 ‘cfv 5131 1^st c1st 6044 2^nd c2nd 6045

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363

This theorem depends on definitions: df-bi 116 df-dc 821 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-1st 6046 df-2nd 6047

This theorem is referenced by: ctiunct 11989

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