ot1stg - Metamath Proof Explorer

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Theorem ot1stg 7854

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 7854, ot2ndg 7855, ot3rdg 7856.) (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 4571	. . . . . 6 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 6786	. . . . 5 ⊢ (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opex 5380	. . . . . 6 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		op1stg 7852	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑋) → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
5	3, 4	mpan 687	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
6	2, 5	eqtrid 2791	. . . 4 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
7	6	fveq2d 6787	. . 3 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (1^st ‘⟨𝐴, 𝐵⟩))
8		op1stg 7852	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
9	7, 8	sylan9eqr 2801	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
10	9	3impa 1109	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 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 Vcvv 3433 ⟨cop 4568 ⟨cotp 4570 ‘cfv 6437 1^st c1st 7838

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-sep 5224 ax-nul 5231 ax-pr 5353 ax-un 7597

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3435 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4258 df-if 4461 df-sn 4563 df-pr 4565 df-op 4569 df-ot 4571 df-uni 4841 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-iota 6395 df-fun 6439 df-fv 6445 df-1st 7840

This theorem is referenced by: oteqimp 7859 el2xptp0 7887 splval 14473 mamufval 21543 msrval 33509 elmsta 33519 mapdhval 39745 hdmap1val 39819

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