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Theorem tpres 6695
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 (𝜑𝑇 = {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
tpres.b (𝜑𝐵𝑉)
tpres.c (𝜑𝐶𝑉)
tpres.e (𝜑𝐸𝑉)
tpres.f (𝜑𝐹𝑉)
tpres.1 (𝜑𝐵𝐴)
tpres.2 (𝜑𝐶𝐴)
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.
StepHypRef Expression
1 df-res 5324 . 2 (𝑇 ↾ (V ∖ {𝐴})) = (𝑇 ∩ ((V ∖ {𝐴}) × V))
2 elin 3994 . . . 4 (𝑥 ∈ (𝑇 ∩ ((V ∖ {𝐴}) × V)) ↔ (𝑥𝑇𝑥 ∈ ((V ∖ {𝐴}) × V)))
3 elxp 5335 . . . . . 6 (𝑥 ∈ ((V ∖ {𝐴}) × V) ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))
43anbi2i 617 . . . . 5 ((𝑥𝑇𝑥 ∈ ((V ∖ {𝐴}) × V)) ↔ (𝑥𝑇 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))))
5 tpres.t . . . . . . . . 9 (𝜑𝑇 = {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
65eleq2d 2864 . . . . . . . 8 (𝜑 → (𝑥𝑇𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
7 vex 3388 . . . . . . . . . . . . 13 𝑥 ∈ V
87eltp 4420 . . . . . . . . . . . 12 (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} ↔ (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
9 eldifsn 4506 . . . . . . . . . . . . . . . . . . . 20 (𝑎 ∈ (V ∖ {𝐴}) ↔ (𝑎 ∈ V ∧ 𝑎𝐴))
10 eqeq1 2803 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥 = ⟨𝑎, 𝑏⟩ → (𝑥 = ⟨𝐴, 𝐷⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩))
1110adantl 474 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥 = ⟨𝑎, 𝑏⟩) → (𝑥 = ⟨𝐴, 𝐷⟩ ↔ ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩))
12 vex 3388 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑎 ∈ V
13 vex 3388 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑏 ∈ V
1412, 13opth 5135 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩ ↔ (𝑎 = 𝐴𝑏 = 𝐷))
15 eqneqall 2982 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑎 = 𝐴 → (𝑎𝐴 → (𝑏 = 𝐷 → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))))
1615com12 32 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑎𝐴 → (𝑎 = 𝐴 → (𝑏 = 𝐷 → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))))
1716impd 399 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑎𝐴 → ((𝑎 = 𝐴𝑏 = 𝐷) → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
1814, 17syl5bi 234 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑎𝐴 → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩ → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
1918adantr 473 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐴𝑥 = ⟨𝑎, 𝑏⟩) → (⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐷⟩ → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
2011, 19sylbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑎𝐴𝑥 = ⟨𝑎, 𝑏⟩) → (𝑥 = ⟨𝐴, 𝐷⟩ → (𝜑 → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
2120impd 399 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑎𝐴𝑥 = ⟨𝑎, 𝑏⟩) → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
2221ex 402 . . . . . . . . . . . . . . . . . . . . 21 (𝑎𝐴 → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
2322adantl 474 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 ∈ V ∧ 𝑎𝐴) → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
249, 23sylbi 209 . . . . . . . . . . . . . . . . . . 19 (𝑎 ∈ (V ∖ {𝐴}) → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
2524adantr 473 . . . . . . . . . . . . . . . . . 18 ((𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V) → (𝑥 = ⟨𝑎, 𝑏⟩ → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
2625impcom 397 . . . . . . . . . . . . . . . . 17 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
2726com12 32 . . . . . . . . . . . . . . . 16 ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
2827exlimdvv 2030 . . . . . . . . . . . . . . 15 ((𝑥 = ⟨𝐴, 𝐷⟩ ∧ 𝜑) → (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
2928ex 402 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝐴, 𝐷⟩ → (𝜑 → (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))))
3029impd 399 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐴, 𝐷⟩ → ((𝜑 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
31 orc 894 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝐵, 𝐸⟩ → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
3231a1d 25 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐵, 𝐸⟩ → ((𝜑 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
33 olc 895 . . . . . . . . . . . . . 14 (𝑥 = ⟨𝐶, 𝐹⟩ → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
3433a1d 25 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐶, 𝐹⟩ → ((𝜑 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
3530, 32, 343jaoi 1553 . . . . . . . . . . . 12 ((𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) → ((𝜑 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
368, 35sylbi 209 . . . . . . . . . . 11 (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → ((𝜑 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
377elpr 4391 . . . . . . . . . . 11 (𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} ↔ (𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
3836, 37syl6ibr 244 . . . . . . . . . 10 (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → ((𝜑 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
3938expd 405 . . . . . . . . 9 (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → (𝜑 → (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})))
4039com12 32 . . . . . . . 8 (𝜑 → (𝑥 ∈ {⟨𝐴, 𝐷⟩, ⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})))
416, 40sylbid 232 . . . . . . 7 (𝜑 → (𝑥𝑇 → (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})))
4241impd 399 . . . . . 6 (𝜑 → ((𝑥𝑇 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) → 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
43 3mix2 1431 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐵, 𝐸⟩ → (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
44 3mix3 1432 . . . . . . . . . . . . 13 (𝑥 = ⟨𝐶, 𝐹⟩ → (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
4543, 44jaoi 884 . . . . . . . . . . . 12 ((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) → (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
4645adantr 473 . . . . . . . . . . 11 (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩))
476, 8syl6bb 279 . . . . . . . . . . . 12 (𝜑 → (𝑥𝑇 ↔ (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
4847adantl 474 . . . . . . . . . . 11 (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → (𝑥𝑇 ↔ (𝑥 = ⟨𝐴, 𝐷⟩ ∨ 𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩)))
4946, 48mpbird 249 . . . . . . . . . 10 (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → 𝑥𝑇)
50 tpres.b . . . . . . . . . . . . . . . 16 (𝜑𝐵𝑉)
51 elex 3400 . . . . . . . . . . . . . . . 16 (𝐵𝑉𝐵 ∈ V)
5250, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐵 ∈ V)
53 tpres.1 . . . . . . . . . . . . . . 15 (𝜑𝐵𝐴)
54 tpres.e . . . . . . . . . . . . . . . 16 (𝜑𝐸𝑉)
55 elex 3400 . . . . . . . . . . . . . . . 16 (𝐸𝑉𝐸 ∈ V)
5654, 55syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐸 ∈ V)
5752, 53, 56jca31 511 . . . . . . . . . . . . . 14 (𝜑 → ((𝐵 ∈ V ∧ 𝐵𝐴) ∧ 𝐸 ∈ V))
5857anim2i 611 . . . . . . . . . . . . 13 ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ 𝜑) → (𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵𝐴) ∧ 𝐸 ∈ V)))
59 opeq12 4595 . . . . . . . . . . . . . . . . . 18 ((𝑎 = 𝐵𝑏 = 𝐸) → ⟨𝑎, 𝑏⟩ = ⟨𝐵, 𝐸⟩)
6059eqeq2d 2809 . . . . . . . . . . . . . . . . 17 ((𝑎 = 𝐵𝑏 = 𝐸) → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐵, 𝐸⟩))
61 eleq1 2866 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝐵 → (𝑎 ∈ V ↔ 𝐵 ∈ V))
62 neeq1 3033 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝐵 → (𝑎𝐴𝐵𝐴))
6361, 62anbi12d 625 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝐵 → ((𝑎 ∈ V ∧ 𝑎𝐴) ↔ (𝐵 ∈ V ∧ 𝐵𝐴)))
64 eleq1 2866 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝐸 → (𝑏 ∈ V ↔ 𝐸 ∈ V))
6563, 64bi2anan9 630 . . . . . . . . . . . . . . . . 17 ((𝑎 = 𝐵𝑏 = 𝐸) → (((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V) ↔ ((𝐵 ∈ V ∧ 𝐵𝐴) ∧ 𝐸 ∈ V)))
6660, 65anbi12d 625 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝐵𝑏 = 𝐸) → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V)) ↔ (𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵𝐴) ∧ 𝐸 ∈ V))))
6766spc2egv 3483 . . . . . . . . . . . . . . 15 ((𝐵𝑉𝐸𝑉) → ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵𝐴) ∧ 𝐸 ∈ V)) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V))))
6850, 54, 67syl2anc 580 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵𝐴) ∧ 𝐸 ∈ V)) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V))))
6968adantl 474 . . . . . . . . . . . . 13 ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ 𝜑) → ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ ((𝐵 ∈ V ∧ 𝐵𝐴) ∧ 𝐸 ∈ V)) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V))))
7058, 69mpd 15 . . . . . . . . . . . 12 ((𝑥 = ⟨𝐵, 𝐸⟩ ∧ 𝜑) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V)))
71 tpres.c . . . . . . . . . . . . . . . 16 (𝜑𝐶𝑉)
72 elex 3400 . . . . . . . . . . . . . . . 16 (𝐶𝑉𝐶 ∈ V)
7371, 72syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐶 ∈ V)
74 tpres.2 . . . . . . . . . . . . . . 15 (𝜑𝐶𝐴)
75 tpres.f . . . . . . . . . . . . . . . 16 (𝜑𝐹𝑉)
76 elex 3400 . . . . . . . . . . . . . . . 16 (𝐹𝑉𝐹 ∈ V)
7775, 76syl 17 . . . . . . . . . . . . . . 15 (𝜑𝐹 ∈ V)
7873, 74, 77jca31 511 . . . . . . . . . . . . . 14 (𝜑 → ((𝐶 ∈ V ∧ 𝐶𝐴) ∧ 𝐹 ∈ V))
7978anim2i 611 . . . . . . . . . . . . 13 ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ 𝜑) → (𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶𝐴) ∧ 𝐹 ∈ V)))
80 opeq12 4595 . . . . . . . . . . . . . . . . . 18 ((𝑎 = 𝐶𝑏 = 𝐹) → ⟨𝑎, 𝑏⟩ = ⟨𝐶, 𝐹⟩)
8180eqeq2d 2809 . . . . . . . . . . . . . . . . 17 ((𝑎 = 𝐶𝑏 = 𝐹) → (𝑥 = ⟨𝑎, 𝑏⟩ ↔ 𝑥 = ⟨𝐶, 𝐹⟩))
82 eleq1 2866 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝐶 → (𝑎 ∈ V ↔ 𝐶 ∈ V))
83 neeq1 3033 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝐶 → (𝑎𝐴𝐶𝐴))
8482, 83anbi12d 625 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝐶 → ((𝑎 ∈ V ∧ 𝑎𝐴) ↔ (𝐶 ∈ V ∧ 𝐶𝐴)))
85 eleq1 2866 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝐹 → (𝑏 ∈ V ↔ 𝐹 ∈ V))
8684, 85bi2anan9 630 . . . . . . . . . . . . . . . . 17 ((𝑎 = 𝐶𝑏 = 𝐹) → (((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V) ↔ ((𝐶 ∈ V ∧ 𝐶𝐴) ∧ 𝐹 ∈ V)))
8781, 86anbi12d 625 . . . . . . . . . . . . . . . 16 ((𝑎 = 𝐶𝑏 = 𝐹) → ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V)) ↔ (𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶𝐴) ∧ 𝐹 ∈ V))))
8887spc2egv 3483 . . . . . . . . . . . . . . 15 ((𝐶𝑉𝐹𝑉) → ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶𝐴) ∧ 𝐹 ∈ V)) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V))))
8971, 75, 88syl2anc 580 . . . . . . . . . . . . . 14 (𝜑 → ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶𝐴) ∧ 𝐹 ∈ V)) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V))))
9089adantl 474 . . . . . . . . . . . . 13 ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ 𝜑) → ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ ((𝐶 ∈ V ∧ 𝐶𝐴) ∧ 𝐹 ∈ V)) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V))))
9179, 90mpd 15 . . . . . . . . . . . 12 ((𝑥 = ⟨𝐶, 𝐹⟩ ∧ 𝜑) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V)))
9270, 91jaoian 980 . . . . . . . . . . 11 (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V)))
939anbi1i 618 . . . . . . . . . . . . 13 ((𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V) ↔ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V))
9493anbi2i 617 . . . . . . . . . . . 12 ((𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) ↔ (𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V)))
95942exbii 1945 . . . . . . . . . . 11 (∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)) ↔ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ ((𝑎 ∈ V ∧ 𝑎𝐴) ∧ 𝑏 ∈ V)))
9692, 95sylibr 226 . . . . . . . . . 10 (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))
9749, 96jca 508 . . . . . . . . 9 (((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) ∧ 𝜑) → (𝑥𝑇 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))))
9897ex 402 . . . . . . . 8 ((𝑥 = ⟨𝐵, 𝐸⟩ ∨ 𝑥 = ⟨𝐶, 𝐹⟩) → (𝜑 → (𝑥𝑇 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))))
9937, 98sylbi 209 . . . . . . 7 (𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → (𝜑 → (𝑥𝑇 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))))
10099com12 32 . . . . . 6 (𝜑 → (𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩} → (𝑥𝑇 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V)))))
10142, 100impbid 204 . . . . 5 (𝜑 → ((𝑥𝑇 ∧ ∃𝑎𝑏(𝑥 = ⟨𝑎, 𝑏⟩ ∧ (𝑎 ∈ (V ∖ {𝐴}) ∧ 𝑏 ∈ V))) ↔ 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
1024, 101syl5bb 275 . . . 4 (𝜑 → ((𝑥𝑇𝑥 ∈ ((V ∖ {𝐴}) × V)) ↔ 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
1032, 102syl5bb 275 . . 3 (𝜑 → (𝑥 ∈ (𝑇 ∩ ((V ∖ {𝐴}) × V)) ↔ 𝑥 ∈ {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩}))
104103eqrdv 2797 . 2 (𝜑 → (𝑇 ∩ ((V ∖ {𝐴}) × V)) = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
1051, 104syl5eq 2845 1 (𝜑 → (𝑇 ↾ (V ∖ {𝐴})) = {⟨𝐵, 𝐸⟩, ⟨𝐶, 𝐹⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wo 874  w3o 1107   = wceq 1653  wex 1875  wcel 2157  wne 2971  Vcvv 3385  cdif 3766  cin 3768  {csn 4368  {cpr 4370  {ctp 4372  cop 4374   × cxp 5310  cres 5314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-opab 4906  df-xp 5318  df-res 5324
This theorem is referenced by:  estrresOLD  17093  estrres  17094
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