ot1stg - Metamath Proof Explorer

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Theorem ot1stg 7945

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 7945, ot2ndg 7946, ot3rdg 7947.) (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 4566	. . . . . 6 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 6832	. . . . 5 ⊢ (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opex 5405	. . . . . 6 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		op1stg 7943	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑋) → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
5	3, 4	mpan 691	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
6	2, 5	eqtrid 2782	. . . 4 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
7	6	fveq2d 6833	. . 3 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (1^st ‘⟨𝐴, 𝐵⟩))
8		op1stg 7943	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
9	7, 8	sylan9eqr 2792	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
10	9	3impa 1110	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3427 ⟨cop 4563 ⟨cotp 4565 ‘cfv 6487 1^st c1st 7929

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 2184 ax-ext 2707 ax-sep 5220 ax-nul 5230 ax-pr 5364 ax-un 7678

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-ral 3050 df-rex 3060 df-rab 3388 df-v 3429 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4264 df-if 4457 df-sn 4558 df-pr 4560 df-op 4564 df-ot 4566 df-uni 4841 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-iota 6443 df-fun 6489 df-fv 6495 df-1st 7931

This theorem is referenced by: oteqimp 7950 el2xptp0 7978 sbcoteq1a 7993 xpord3lem 8088 splval 14702 mamufval 22345 msrval 35708 elmsta 35718 mapdhval 42158 hdmap1val 42232

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