ot1stg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ot1stg		Structured version Visualization version GIF version

Theorem ot1stg 7985

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 7985, ot2ndg 7986, ot3rdg 7987.) (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 4636	. . . . . 6 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 6891	. . . . 5 ⊢ (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opex 5463	. . . . . 6 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
4		op1stg 7983	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑋) → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
5	3, 4	mpan 688	. . . . 5 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
6	2, 5	eqtrid 2784	. . . 4 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
7	6	fveq2d 6892	. . 3 ⊢ (𝐶 ∈ 𝑋 → (1^st ‘(1^st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (1^st ‘⟨𝐴, 𝐵⟩))
8		op1stg 7983	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1^st ‘⟨𝐴, 𝐵⟩) = 𝐴)
9	7, 8	sylan9eqr 2794	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟨cop 4633 ⟨cotp 4635 ‘cfv 6540 1^st c1st 7969

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-ot 4636 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6492 df-fun 6542 df-fv 6548 df-1st 7971

This theorem is referenced by: oteqimp 7990 el2xptp0 8018 sbcoteq1a 8033 xpord3lem 8131 splval 14697 mamufval 21878 msrval 34517 elmsta 34527 mapdhval 40583 hdmap1val 40657

W3C validator