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Theorem oteqimp 7940

Description: The components of an ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)

**Assertion**
Ref	Expression
oteqimp	⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = 𝐵 ∧ (2^nd ‘𝑇) = 𝐶)))

**Proof of Theorem oteqimp**
Step	Hyp	Ref	Expression
1		ot1stg 7935	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2		ot2ndg 7936	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3		ot3rdg 7937	. . . 4 ⊢ (𝐶 ∈ 𝑍 → (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
4	3	3ad2ant3 1135	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
5	1, 2, 4	3jca 1128	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴 ∧ (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵 ∧ (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
6		2fveq3 6827	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1^st ‘(1^st ‘𝑇)) = (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
7	6	eqeq1d 2733	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ↔ (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
8		2fveq3 6827	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2^nd ‘(1^st ‘𝑇)) = (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
9	8	eqeq1d 2733	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘(1^st ‘𝑇)) = 𝐵 ↔ (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
10		fveqeq2 6831	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘𝑇) = 𝐶 ↔ (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
11	7, 9, 10	3anbi123d 1438	. 2 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = 𝐵 ∧ (2^nd ‘𝑇) = 𝐶) ↔ ((1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴 ∧ (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵 ∧ (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)))
12	5, 11	imbitrrid 246	1 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ∧ (2^nd ‘(1^st ‘𝑇)) = 𝐵 ∧ (2^nd ‘𝑇) = 𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ⟨cotp 4584 ‘cfv 6481 1^st c1st 7919 2^nd c2nd 7920

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5234 ax-nul 5244 ax-pr 5370 ax-un 7668

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4476 df-sn 4577 df-pr 4579 df-op 4583 df-ot 4585 df-uni 4860 df-br 5092 df-opab 5154 df-mpt 5173 df-id 5511 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-iota 6437 df-fun 6483 df-fv 6489 df-1st 7921 df-2nd 7922

This theorem is referenced by: (None)

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