Theorem ot1stg 6043

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 6043, ot2ndg 6044, ot3rdgg 6045.) (Contributed by NM, 3-Apr-2015.) (Revised by Mario Carneiro, 2-May-2015.)

Assertion

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

Proof of Theorem ot1stg

Step	Hyp	Ref	Expression
1		df-ot 3532	. . . . 5 ⊢ ⟨𝐴, 𝐵, 𝐶⟩ = ⟨⟨𝐴, 𝐵⟩, 𝐶⟩
2	1	fveq2i 5417	. . . 4 ⊢ (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩)
3		opexg 4145	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ V)
4		op1stg 6041	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ 𝐶 ∈ 𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
5	3, 4	sylan 281	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐶 ∈ 𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
6	5	3impa 1176	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘⟨⟨𝐴, 𝐵⟩, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
7	2, 6	syl5eq 2182	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘⟨𝐴, 𝐵, 𝐶⟩) = ⟨𝐴, 𝐵⟩)
8	7	fveq2d 5418	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = (1st ‘⟨𝐴, 𝐵⟩))
9		op1stg 6041	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
10	9	3adant3 1001	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
11	8, 10	eqtrd 2170	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (1st ‘(1st ‘⟨𝐴, 𝐵, 𝐶⟩)) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  Vcvv 2681  ⟨cop 3525  ⟨cotp 3526  ‘cfv 5118  1^st c1st 6029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-ot 3532  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-1st 6031

This theorem is referenced by: (None)