Theorem op1stg 7986

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 4873	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6895	. . 3 ⊢ (𝑥 = 𝐴 → (1st ‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝐴, 𝑦⟩))
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2748	. 2 ⊢ (𝑥 = 𝐴 → ((1st ‘⟨𝑥, 𝑦⟩) = 𝑥 ↔ (1st ‘⟨𝐴, 𝑦⟩) = 𝐴))
5		opeq2 4874	. . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
6	5	fveqeq2d 6899	. 2 ⊢ (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴))
7		vex 3478	. . 3 ⊢ 𝑥 ∈ V
8		vex 3478	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op1st 7982	. 2 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
10	4, 6, 9	vtocl2g 3562	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ⟨cop 4634  ‘cfv 6543  1^st c1st 7972

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 5299  ax-nul 5306  ax-pr 5427  ax-un 7724

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 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fv 6551  df-1st 7974

This theorem is referenced by:  ot1stg  7988  ot2ndg  7989  br1steqg  7996  1stconst  8085  mposn  8088  curry2  8092  opco1  8108  mpoxopn0yelv  8197  mpoxopoveq  8203  xpmapenlem  9143  1stinl  9921  1stinr  9923  fpwwe  10640  addpipq  10931  mulpipq  10934  ordpipq  10936  swrdval  14592  ruclem1  16173  qnumdenbi  16679  setsstruct  17108  oppccofval  17660  funcf2  17817  cofuval2  17836  resfval2  17842  resf1st  17843  isnat  17897  fucco  17914  homadm  17989  setcco  18032  estrcco  18080  xpcco  18134  xpchom2  18137  xpcco2  18138  evlf2  18170  curfval  18175  curf1cl  18180  uncf1  18188  uncf2  18189  diag11  18195  diag12  18196  diag2  18197  hof2fval  18207  yonedalem21  18225  yonedalem22  18230  mvmulfval  22043  imasdsf1olem  23878  ovolicc1  25032  ioombl1lem3  25076  ioombl1lem4  25077  addsqnreup  26943  addsval  27443  mulsval  27562  brcgr  28155  opvtxfv  28261  fgreu  31892  fsuppcurry2  31946  sategoelfvb  34405  prv1n  34417  fvtransport  34999  bj-inftyexpiinv  36084  bj-finsumval0  36161  poimirlem17  36500  poimirlem24  36507  poimirlem27  36510  rngoablo2  36772  dvhopvadd  39959  dvhopvsca  39968  dvhopaddN  39980  dvhopspN  39981  etransclem44  44984  ovnsubaddlem1  45276  ovnlecvr2  45316  ovolval5lem2  45359  rngccoALTV  46876  ringccoALTV  46939