Theorem tpres 7206

Description: An unordered triple of ordered pairs restricted to all but one first components of the pairs is an unordered pair of ordered pairs. (Contributed by AV, 14-Mar-2020.)

Hypotheses

Ref	Expression
tpres.t	⊢ (𝜑 → 𝑇 = {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
tpres.b	⊢ (𝜑 → 𝐵 ∈ 𝑉)
tpres.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
tpres.e	⊢ (𝜑 → 𝐸 ∈ 𝑉)
tpres.f	⊢ (𝜑 → 𝐹 ∈ 𝑉)
tpres.1	⊢ (𝜑 → 𝐵 ≠ 𝐴)
tpres.2	⊢ (𝜑 → 𝐶 ≠ 𝐴)

Assertion

Ref	Expression
tpres	⊢ (𝜑 → (𝑇 ↾ (V ∖ {𝐴})) = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})

Proof of Theorem tpres

Dummy variables 𝑎 𝑏 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-res 5675	. 2 ⊢ (𝑇 ↾ (V ∖ {𝐴})) = (𝑇 ∩ ((V ∖ {𝐴}) × V))
2		elin 3922	. . . 4 ⊢ (𝑥 ∈ (𝑇 ∩ ((V ∖ {𝐴}) × V)) ↔ (𝑥 ∈ 𝑇 ∧ 𝑥 ∈ ((V ∖ {𝐴}) × V)))
3		elxp 5686	. . . . . 6 ⊢ (𝑥 ∈ ((V ∖ {𝐴}) × V) ↔ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))
4	3	anbi2i 635	. . . . 5 ⊢ ((𝑥 ∈ 𝑇 ∧ 𝑥 ∈ ((V ∖ {𝐴}) × V)) ↔ (𝑥 ∈ 𝑇 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))))
5		tpres.t	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 = {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
6	5	eleq2d 2851	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ 𝑇 ↔ 𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
7		vex 3461	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
8	7	eltp 4657	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} ↔ (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
9		eldifsn 4755	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ (V ∖ {𝐴}) ↔ (𝑎 ∈ V ∧ 𝑎 ≠ 𝐴))
10		eqeq1 2769	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 = ⟨𝐴, 𝐷⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩))
11	10	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ≠ 𝐴 ∧ 𝑥 = ⟨𝑎, 𝑏⟩) → (𝑥 = ⟨𝐴, 𝐷⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩))
12		vex 3461	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑎 ∈ V
13		vex 3461	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑏 ∈ V
14	12, 13	opth 5460	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩ ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐷))
15		eqneqall 2971	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑎 = 𝐴 → (𝑎 ≠ 𝐴 → (𝑏 = 𝐷 → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))))
16	15	com12 33	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑎 ≠ 𝐴 → (𝑎 = 𝐴 → (𝑏 = 𝐷 → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))))
17	16	impd 416	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑎 ≠ 𝐴 → ((𝑎 = 𝐴 ∧ 𝑏 = 𝐷) → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
18	14, 17	biimtrid 245	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑎 ≠ 𝐴 → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩ → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
19	18	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ≠ 𝐴 ∧ 𝑥 = ⟨𝑎, 𝑏⟩) → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩ → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
20	11, 19	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑎 ≠ 𝐴 ∧ 𝑥 = ⟨𝑎, 𝑏⟩) → (𝑥 = ⟨𝐴, 𝐷⟩ → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
21	20	impd 416	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑎 ≠ 𝐴 ∧ 𝑥 = ⟨𝑎, 𝑏⟩) → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
22	21	ex 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 ≠ 𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
23	22	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
24	9, 23	sylbi 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 ∈ (V ∖ {𝐴}) → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
25	24	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V) → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
26	25	impcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
27	26	com12 33	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
28	27	exlimdvv 1967	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
29	28	ex 418	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝐴, 𝐷⟩ → (𝜑 → (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
30	29	impd 416	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝐴, 𝐷⟩ → ((𝜑 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
31		orc 881	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝐵, 𝐸⟩ → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
32	31	a1d 26	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝐵, 𝐸⟩ → ((𝜑 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
33		olc 882	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨𝐶, 𝐹⟩ → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
34	33	a1d 26	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝐶, 𝐹⟩ → ((𝜑 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
35	30, 32, 34	3jaoi 1454	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) → ((𝜑 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
36	8, 35	sylbi 220	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → ((𝜑 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
37	7	elpr 4616	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} ↔ (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
38	36, 37	imbitrrdi 255	. . . . . . . . . 10 ⊢ (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → ((𝜑 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
39	38	expd 421	. . . . . . . . 9 ⊢ (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → (𝜑 → (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})))
40	39	com12 33	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})))
41	6, 40	sylbid 243	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑇 → (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})))
42	41	impd 416	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝑇 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
43		3mix2 1350	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝐵, 𝐸⟩ → (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
44		3mix3 1351	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨𝐶, 𝐹⟩ → (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
45	43, 44	jaoi 871	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) → (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
46	45	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
47	6, 8	bitrdi 290	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑥 ∈ 𝑇 ↔ (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
48	47	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → (𝑥 ∈ 𝑇 ↔ (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
49	46, 48	mpbird 260	. . . . . . . . . 10 ⊢ (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → 𝑥 ∈ 𝑇)
50		tpres.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
51	50	elexd 3480	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ V)
52		tpres.1	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ≠ 𝐴)
53		tpres.e	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
54	53	elexd 3480	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐸 ∈ V)
55	51, 52, 54	jca31 524	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐵 ∈ V ∧ 𝐵 ≠ 𝐴) ∧ 𝐸 ∈ V))
56	55	anim2i 629	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵 ≠ 𝐴) ∧ 𝐸 ∈ V)))
57		opeq12 4842	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐸) → ⟨𝑎, 𝑏⟩ = ⟨𝐵, 𝐸⟩)
58	57	eqeq2d 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐸) → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, 𝐸⟩))
59		eleq1 2853	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝐵 → (𝑎 ∈ V ↔ 𝐵 ∈ V))
60		neeq1 3022	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝐵 → (𝑎 ≠ 𝐴 ↔ 𝐵 ≠ 𝐴))
61	59, 60	anbi12d 644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝐵 → ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ↔ (𝐵 ∈ V ∧ 𝐵 ≠ 𝐴)))
62		eleq1 2853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐸 → (𝑏 ∈ V ↔ 𝐸 ∈ V))
63	61, 62	bi2anan9 650	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐸) → (((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V) ↔ ((𝐵 ∈ V ∧ 𝐵 ≠ 𝐴) ∧ 𝐸 ∈ V)))
64	58, 63	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝐵 ∧ 𝑏 = 𝐸) → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V)) ↔ (𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵 ≠ 𝐴) ∧ 𝐸 ∈ V))))
65	64	spc2egv 3560	. . . . . . . . . . . . . . 15 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐸 ∈ 𝑉) → ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵 ≠ 𝐴) ∧ 𝐸 ∈ V)) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V))))
66	50, 53, 65	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵 ≠ 𝐴) ∧ 𝐸 ∈ V)) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V))))
67	66	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ 𝜑) → ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵 ≠ 𝐴) ∧ 𝐸 ∈ V)) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V))))
68	56, 67	mpd 16	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ 𝜑) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V)))
69		tpres.c	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
70	69	elexd 3480	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ V)
71		tpres.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ≠ 𝐴)
72		tpres.f	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 ∈ 𝑉)
73	72	elexd 3480	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ V)
74	70, 71, 73	jca31 524	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ∧ 𝐹 ∈ V))
75	74	anim2i 629	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ 𝜑) → (𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ∧ 𝐹 ∈ V)))
76		opeq12 4842	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 = 𝐶 ∧ 𝑏 = 𝐹) → ⟨𝑎, 𝑏⟩ = ⟨𝐶, 𝐹⟩)
77	76	eqeq2d 2776	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝐶 ∧ 𝑏 = 𝐹) → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐶, 𝐹⟩))
78		eleq1 2853	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ V ↔ 𝐶 ∈ V))
79		neeq1 3022	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝐶 → (𝑎 ≠ 𝐴 ↔ 𝐶 ≠ 𝐴))
80	78, 79	anbi12d 644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝐶 → ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ↔ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐴)))
81		eleq1 2853	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 = 𝐹 → (𝑏 ∈ V ↔ 𝐹 ∈ V))
82	80, 81	bi2anan9 650	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 = 𝐶 ∧ 𝑏 = 𝐹) → (((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V) ↔ ((𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ∧ 𝐹 ∈ V)))
83	77, 82	anbi12d 644	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 = 𝐶 ∧ 𝑏 = 𝐹) → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V)) ↔ (𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ∧ 𝐹 ∈ V))))
84	83	spc2egv 3560	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐹 ∈ 𝑉) → ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ∧ 𝐹 ∈ V)) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V))))
85	69, 72, 84	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ∧ 𝐹 ∈ V)) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V))))
86	85	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ 𝜑) → ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶 ≠ 𝐴) ∧ 𝐹 ∈ V)) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V))))
87	75, 86	mpd 16	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ 𝜑) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V)))
88	68, 87	jaoian 971	. . . . . . . . . . 11 ⊢ (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V)))
89	9	anbi1i 636	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V) ↔ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V))
90	89	anbi2i 635	. . . . . . . . . . . 12 ⊢ ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V)))
91	90	2exbii 1882	. . . . . . . . . . 11 ⊢ (∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) ↔ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎 ≠ 𝐴) ∧ 𝑏 ∈ V)))
92	88, 91	sylibr 237	. . . . . . . . . 10 ⊢ (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))
93	49, 92	jca 521	. . . . . . . . 9 ⊢ (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → (𝑥 ∈ 𝑇 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))))
94	93	ex 418	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) → (𝜑 → (𝑥 ∈ 𝑇 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))))
95	37, 94	sylbi 220	. . . . . . 7 ⊢ (𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → (𝜑 → (𝑥 ∈ 𝑇 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))))
96	95	com12 33	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → (𝑥 ∈ 𝑇 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))))
97	42, 96	impbid 215	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑇 ∧ ∃𝑎∃𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) ↔ 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
98	4, 97	bitrid 286	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑇 ∧ 𝑥 ∈ ((V ∖ {𝐴}) × V)) ↔ 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
99	2, 98	bitrid 286	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝑇 ∩ ((V ∖ {𝐴}) × V)) ↔ 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
100	99	eqrdv 2763	. 2 ⊢ (𝜑 → (𝑇 ∩ ((V ∖ {𝐴}) × V)) = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
101	1, 100	eqtrid 2812	1 ⊢ (𝜑 → (𝑇 ↾ (V ∖ {𝐴})) = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  Vcvv 3457   ∖ cdif 3903   ∩ cin 3905  {csn 4591  {cpr 4593  {ctp 4595  ⟨cop 4597   × cxp 5661   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-opab 5176  df-xp 5669  df-res 5675

This theorem is used by:  estrres  18219  symgvalstruct  19513