Theorem op1stg 6048

Description: Extract the first member of an ordered pair. (Contributed by NM, 19-Jul-2005.)

Assertion

Ref	Expression
op1stg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Proof of Theorem op1stg

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq1 3705	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 5425	. . 3 ⊢ (𝑥 = 𝐴 → (1st ‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝐴, 𝑦⟩))
3		id 19	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2154	. 2 ⊢ (𝑥 = 𝐴 → ((1st ‘⟨𝑥, 𝑦⟩) = 𝑥 ↔ (1st ‘⟨𝐴, 𝑦⟩) = 𝐴))
5		opeq2 3706	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
6	5	fveq2d 5425	. . 3 ⊢ (𝑦 = 𝐵 → (1st ‘⟨𝐴, 𝑦⟩) = (1st ‘⟨𝐴, 𝐵⟩))
7	6	eqeq1d 2148	. 2 ⊢ (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴))
8		vex 2689	. . 3 ⊢ 𝑥 ∈ V
9		vex 2689	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	op1st 6044	. 2 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
11	4, 7, 10	vtocl2g 2750	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ⟨cop 3530  ‘cfv 5123  1^st c1st 6036

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-1st 6038

This theorem is referenced by:  ot1stg  6050  ot2ndg  6051  1stconst  6118  algrflemg  6127  mpoxopn0yelv  6136  mpoxopoveq  6137  xpmapenlem  6743  1stinl  6959  1stinr  6961  mulpipq  7180  suplocexprlemlub  7532  aprcl  8408  frecuzrdgg  10189  qredeu  11778  qnumdenbi  11870  upxp  12441  uptx  12443  txmetcnp  12687