Theorem op1stg 7751

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 4770	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6699	. . 3 ⊢ (𝑥 = 𝐴 → (1st ‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝐴, 𝑦⟩))
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2752	. 2 ⊢ (𝑥 = 𝐴 → ((1st ‘⟨𝑥, 𝑦⟩) = 𝑥 ↔ (1st ‘⟨𝐴, 𝑦⟩) = 𝐴))
5		opeq2 4771	. . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
6	5	fveqeq2d 6703	. 2 ⊢ (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴))
7		vex 3402	. . 3 ⊢ 𝑥 ∈ V
8		vex 3402	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op1st 7747	. 2 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
10	4, 6, 9	vtocl2g 3476	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ⟨cop 4533  ‘cfv 6358  1^st c1st 7737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-iota 6316  df-fun 6360  df-fv 6366  df-1st 7739

This theorem is referenced by:  ot1stg  7753  ot2ndg  7754  br1steqg  7761  1stconst  7846  mposn  7849  curry2  7853  mpoxopn0yelv  7933  mpoxopoveq  7939  xpmapenlem  8791  1stinl  9508  1stinr  9510  fpwwe  10225  addpipq  10516  mulpipq  10519  ordpipq  10521  swrdval  14173  ruclem1  15755  qnumdenbi  16263  setsstruct  16705  oppccofval  17174  funcf2  17328  cofuval2  17347  resfval2  17353  resf1st  17354  isnat  17408  fucco  17425  homadm  17500  setcco  17543  estrcco  17591  xpcco  17644  xpchom2  17647  xpcco2  17648  evlf2  17680  curfval  17685  curf1cl  17690  uncf1  17698  uncf2  17699  diag11  17705  diag12  17706  diag2  17707  hof2fval  17717  yonedalem21  17735  yonedalem22  17740  mvmulfval  21393  imasdsf1olem  23225  ovolicc1  24367  ioombl1lem3  24411  ioombl1lem4  24412  addsqnreup  26278  brcgr  26945  opvtxfv  27049  fgreu  30683  fsuppcurry2  30735  sategoelfvb  33048  prv1n  33060  addsval  33812  fvtransport  34020  bj-inftyexpiinv  35063  bj-finsumval0  35140  poimirlem17  35480  poimirlem24  35487  poimirlem27  35490  rngoablo2  35753  dvhopvadd  38793  dvhopvsca  38802  dvhopaddN  38814  dvhopspN  38815  etransclem44  43437  ovnsubaddlem1  43726  ovnlecvr2  43766  ovolval5lem2  43809  rngccoALTV  45162  ringccoALTV  45225