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Theorem oteqimp 8018

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 8013	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2		ot2ndg 8014	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3		ot3rdg 8015	. . . 4 ⊢ (𝐶 ∈ 𝑍 → (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
4	3	3ad2ant3 1132	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
5	1, 2, 4	3jca 1125	. 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 6907	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1^st ‘(1^st ‘𝑇)) = (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
7	6	eqeq1d 2730	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ↔ (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
8		2fveq3 6907	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2^nd ‘(1^st ‘𝑇)) = (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
9	8	eqeq1d 2730	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘(1^st ‘𝑇)) = 𝐵 ↔ (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
10		fveqeq2 6911	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘𝑇) = 𝐶 ↔ (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
11	7, 9, 10	3anbi123d 1432	. 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 1084 = wceq 1533 ∈ wcel 2098 ⟨cotp 4640 ‘cfv 6553 1^st c1st 7997 2^nd c2nd 7998

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7746

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-ot 4641 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fv 6561 df-1st 7999 df-2nd 8000

This theorem is referenced by: (None)

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