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Theorem oteqimp 7993

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 7988	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2		ot2ndg 7989	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3		ot3rdg 7990	. . . 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 6896	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1^st ‘(1^st ‘𝑇)) = (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
7	6	eqeq1d 2734	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ↔ (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
8		2fveq3 6896	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2^nd ‘(1^st ‘𝑇)) = (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
9	8	eqeq1d 2734	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘(1^st ‘𝑇)) = 𝐵 ↔ (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
10		fveqeq2 6900	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘𝑇) = 𝐶 ↔ (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
11	7, 9, 10	3anbi123d 1436	. 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 245	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 1087 = wceq 1541 ∈ wcel 2106 ⟨cotp 4636 ‘cfv 6543 1^st c1st 7972 2^nd c2nd 7973

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-ot 4637 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7974 df-2nd 7975

This theorem is referenced by: (None)

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