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Theorem oteqimp 7962

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 7957	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
2		ot2ndg 7958	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵)
3		ot3rdg 7959	. . . 4 ⊢ (𝐶 ∈ 𝑍 → (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
4	3	3ad2ant3 1136	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑍) → (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶)
5	1, 2, 4	3jca 1129	. 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 6847	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (1^st ‘(1^st ‘𝑇)) = (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
7	6	eqeq1d 2739	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((1^st ‘(1^st ‘𝑇)) = 𝐴 ↔ (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴))
8		2fveq3 6847	. . . 4 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → (2^nd ‘(1^st ‘𝑇)) = (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)))
9	8	eqeq1d 2739	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘(1^st ‘𝑇)) = 𝐵 ↔ (2^nd ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐵))
10		fveqeq2 6851	. . 3 ⊢ (𝑇 = ⟨𝐴, 𝐵, 𝐶⟩ → ((2^nd ‘𝑇) = 𝐶 ↔ (2^nd ‘⟨𝐴, 𝐵, 𝐶⟩) = 𝐶))
11	7, 9, 10	3anbi123d 1439	. 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 1087 = wceq 1542 ∈ wcel 2114 ⟨cotp 4590 ‘cfv 6500 1^st c1st 7941 2^nd c2nd 7942

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-ot 4591 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6456 df-fun 6502 df-fv 6508 df-1st 7943 df-2nd 7944

This theorem is referenced by: (None)

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