ot1stg - Metamath Proof Explorer

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Theorem ot1stg 8003

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 8003, ot2ndg 8004, ot3rdg 8005.) (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 4596	. . . . . 6 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 6885	. . . . 5 ⊢ (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opex 5443	. . . . . 6 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		op1stg 8001	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑋) → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
5	3, 4	mpan 703	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
6	2, 5	eqtrid 2809	. . . 4 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
7	6	fveq2d 6886	. . 3 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (1^st ‘⟨𝐴, 𝐵⟩))
8		op1stg 8001	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
9	7, 8	sylan9eqr 2819	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)
10	9	3impa 1127	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3453 ⟨cop 4593 ⟨cotp 4595 ‘cfv 6537 1^st c1st 7987

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 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pr 5402 ax-un 7739

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-ot 4596 df-uni 4871 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-iota 6493 df-fun 6539 df-fv 6545 df-1st 7989

This theorem is used by: oteqimp 8008 el2xptp0 8036 sbcoteq1a 8051 xpord3lem 8150 splval 14822 mamufval 22615 msrval 36104 elmsta 36114 mapdhval 42584 hdmap1val 42658

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