Theorem op1stg 7987

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 4874	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6896	. . 3 ⊢ (𝑥 = 𝐴 → (1st ‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝐴, 𝑦⟩))
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2749	. 2 ⊢ (𝑥 = 𝐴 → ((1st ‘⟨𝑥, 𝑦⟩) = 𝑥 ↔ (1st ‘⟨𝐴, 𝑦⟩) = 𝐴))
5		opeq2 4875	. . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
6	5	fveqeq2d 6900	. 2 ⊢ (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴))
7		vex 3479	. . 3 ⊢ 𝑥 ∈ V
8		vex 3479	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op1st 7983	. 2 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
10	4, 6, 9	vtocl2g 3563	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ⟨cop 4635  ‘cfv 6544  1^st c1st 7973

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-1st 7975

This theorem is referenced by:  ot1stg  7989  ot2ndg  7990  br1steqg  7997  1stconst  8086  mposn  8089  curry2  8093  opco1  8109  mpoxopn0yelv  8198  mpoxopoveq  8204  xpmapenlem  9144  1stinl  9922  1stinr  9924  fpwwe  10641  addpipq  10932  mulpipq  10935  ordpipq  10937  swrdval  14593  ruclem1  16174  qnumdenbi  16680  setsstruct  17109  oppccofval  17661  funcf2  17818  cofuval2  17837  resfval2  17843  resf1st  17844  isnat  17898  fucco  17915  homadm  17990  setcco  18033  estrcco  18081  xpcco  18135  xpchom2  18138  xpcco2  18139  evlf2  18171  curfval  18176  curf1cl  18181  uncf1  18189  uncf2  18190  diag11  18196  diag12  18197  diag2  18198  hof2fval  18208  yonedalem21  18226  yonedalem22  18231  mvmulfval  22044  imasdsf1olem  23879  ovolicc1  25033  ioombl1lem3  25077  ioombl1lem4  25078  addsqnreup  26946  addsval  27448  mulsval  27568  brcgr  28189  opvtxfv  28295  fgreu  31928  fsuppcurry2  31982  sategoelfvb  34441  prv1n  34453  fvtransport  35035  bj-inftyexpiinv  36137  bj-finsumval0  36214  poimirlem17  36553  poimirlem24  36560  poimirlem27  36563  rngoablo2  36825  dvhopvadd  40012  dvhopvsca  40021  dvhopaddN  40033  dvhopspN  40034  etransclem44  45042  ovnsubaddlem1  45334  ovnlecvr2  45374  ovolval5lem2  45417  rngccoALTV  46934  ringccoALTV  46997