Theorem op1stg 7071

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 4340	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6107	. . 3 ⊢ (𝑥 = 𝐴 → (1st ‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝐴, 𝑦⟩))
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2625	. 2 ⊢ (𝑥 = 𝐴 → ((1st ‘⟨𝑥, 𝑦⟩) = 𝑥 ↔ (1st ‘⟨𝐴, 𝑦⟩) = 𝐴))
5		opeq2 4341	. . . 4 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
6	5	fveq2d 6107	. . 3 ⊢ (𝑦 = 𝐵 → (1st ‘⟨𝐴, 𝑦⟩) = (1st ‘⟨𝐴, 𝐵⟩))
7	6	eqeq1d 2612	. 2 ⊢ (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴))
8		vex 3176	. . 3 ⊢ 𝑥 ∈ V
9		vex 3176	. . 3 ⊢ 𝑦 ∈ V
10	8, 9	op1st 7067	. 2 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
11	4, 7, 10	vtocl2g 3243	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131  ‘cfv 5804  1^st c1st 7057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-1st 7059

This theorem is referenced by:  ot1stg  7073  ot2ndg  7074  1stconst  7152  mpt2sn  7155  curry2  7159  mpt2xopn0yelv  7226  mpt2xopoveq  7232  xpmapenlem  8012  fpwwe  9347  addpipq  9638  mulpipq  9641  ordpipq  9643  swrdval  13269  ruclem1  14799  qnumdenbi  15290  oppccofval  16199  funcf2  16351  cofuval2  16370  resfval2  16376  resf1st  16377  isnat  16430  fucco  16445  homadm  16513  setcco  16556  estrcco  16593  xpcco  16646  xpchom2  16649  xpcco2  16650  evlf2  16681  curfval  16686  curf1cl  16691  uncf1  16699  uncf2  16700  diag11  16706  diag12  16707  diag2  16708  hof2fval  16718  yonedalem21  16736  yonedalem22  16741  mvmulfval  20167  imasdsf1olem  21988  ovolicc1  23091  ioombl1lem3  23135  ioombl1lem4  23136  brcgr  25580  opvtxfv  25681  nbgraop  25952  fgreu  28854  fvtransport  31309  bj-elid3  32264  bj-inftyexpiinv  32272  bj-finsumval0  32324  poimirlem17  32596  poimirlem24  32603  poimirlem27  32606  rngoablo2  32878  dvhopvadd  35400  dvhopvsca  35409  dvhopaddN  35421  dvhopspN  35422  etransclem44  39171  ovnsubaddlem1  39460  ovnlecvr2  39500  ovolval5lem2  39543  rngccoALTV  41780  ringccoALTV  41843