oteqimp - Metamath Proof Explorer

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Theorem oteqimp 8005

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 8000	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2		ot2ndg 8001	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3		ot3rdg 8002	. . . 4 ⊢ (𝐶 ∈ 𝑍 → (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
4	3	3ad2ant3 1153	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
5	1, 2, 4	3jca 1146	. 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 6883	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1^st ‘(1^st ‘𝑇)) = (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
7	6	eqeq1d 2762	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ↔ (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
8		2fveq3 6883	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2^nd ‘(1^st ‘𝑇)) = (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
9	8	eqeq1d 2762	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘(1^st ‘𝑇)) = 𝐵 ↔ (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
10		fveqeq2 6887	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘𝑇) = 𝐶 ↔ (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
11	7, 9, 10	3anbi123d 1464	. 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 249	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

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ⟨cotp 4592 ‘cfv 6533 1^st c1st 7984 2^nd c2nd 7985

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398 ax-un 7736

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-ot 4593 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-iota 6489 df-fun 6535 df-fv 6541 df-1st 7986 df-2nd 7987

This theorem is used by: (None)

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