Theorem op1stg 7843

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 4804	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6778	. . 3 ⊢ (𝑥 = 𝐴 → (1st ‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝐴, 𝑦⟩))
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2754	. 2 ⊢ (𝑥 = 𝐴 → ((1st ‘⟨𝑥, 𝑦⟩) = 𝑥 ↔ (1st ‘⟨𝐴, 𝑦⟩) = 𝐴))
5		opeq2 4805	. . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
6	5	fveqeq2d 6782	. 2 ⊢ (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴))
7		vex 3436	. . 3 ⊢ 𝑥 ∈ V
8		vex 3436	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op1st 7839	. 2 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
10	4, 6, 9	vtocl2g 3510	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ⟨cop 4567  ‘cfv 6433  1^st c1st 7829

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

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 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831

This theorem is referenced by:  ot1stg  7845  ot2ndg  7846  br1steqg  7853  1stconst  7940  mposn  7943  curry2  7947  opco1  7964  mpoxopn0yelv  8029  mpoxopoveq  8035  xpmapenlem  8931  1stinl  9685  1stinr  9687  fpwwe  10402  addpipq  10693  mulpipq  10696  ordpipq  10698  swrdval  14356  ruclem1  15940  qnumdenbi  16448  setsstruct  16877  oppccofval  17426  funcf2  17583  cofuval2  17602  resfval2  17608  resf1st  17609  isnat  17663  fucco  17680  homadm  17755  setcco  17798  estrcco  17846  xpcco  17900  xpchom2  17903  xpcco2  17904  evlf2  17936  curfval  17941  curf1cl  17946  uncf1  17954  uncf2  17955  diag11  17961  diag12  17962  diag2  17963  hof2fval  17973  yonedalem21  17991  yonedalem22  17996  mvmulfval  21691  imasdsf1olem  23526  ovolicc1  24680  ioombl1lem3  24724  ioombl1lem4  24725  addsqnreup  26591  brcgr  27268  opvtxfv  27374  fgreu  31009  fsuppcurry2  31061  sategoelfvb  33381  prv1n  33393  addsval  34126  fvtransport  34334  bj-inftyexpiinv  35379  bj-finsumval0  35456  poimirlem17  35794  poimirlem24  35801  poimirlem27  35804  rngoablo2  36067  dvhopvadd  39107  dvhopvsca  39116  dvhopaddN  39128  dvhopspN  39129  etransclem44  43819  ovnsubaddlem1  44108  ovnlecvr2  44148  ovolval5lem2  44191  rngccoALTV  45546  ringccoALTV  45609