ot1stg - Metamath Proof Explorer

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Theorem ot1stg 7073

Description: Extract the first member of an ordered triple. (Due to infrequent usage, it isn't worthwhile at this point to define special extractors for triples, so we reuse the ordered pair extractors for ot1stg 7073, ot2ndg 7074, ot3rdg 7075.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

**Assertion**
Ref	Expression
ot1stg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)

**Proof of Theorem ot1stg**
Step	Hyp	Ref	Expression
1		df-ot 4134	. . . . . 6 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 6106	. . . . 5 ⊢ (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opex 4859	. . . . . 6 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		op1stg 7071	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑋) → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
5	3, 4	mpan 702	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
6	2, 5	syl5eq 2656	. . . 4 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
7	6	fveq2d 6107	. . 3 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (1^st ‘⟨𝐴, 𝐵⟩))
8		op1stg 7071	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
9	7, 8	sylan9eqr 2666	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
10	9	3impa 1251	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⟨cop 4131 ⟨cotp 4133 ‘cfv 5804 1^st c1st 7057

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-ot 4134 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fv 5812 df-1st 7059

This theorem is referenced by: oteqimp 7078 el2xptp0 7103 splval 13353 mamufval 20010 frg2woteq 26587 msrval 30689 elmsta 30699 mapdhval 36031 hdmap1val 36106

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