Theorem op1stg 7378

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 4559	. . . 4 ⊢ (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
2	1	fveq2d 6379	. . 3 ⊢ (𝑥 = 𝐴 → (1st ‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝐴, 𝑦⟩))
3		id 22	. . 3 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
4	2, 3	eqeq12d 2780	. 2 ⊢ (𝑥 = 𝐴 → ((1st ‘⟨𝑥, 𝑦⟩) = 𝑥 ↔ (1st ‘⟨𝐴, 𝑦⟩) = 𝐴))
5		opeq2 4560	. . 3 ⊢ (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
6	5	fveqeq2d 6383	. 2 ⊢ (𝑦 = 𝐵 → ((1st ‘⟨𝐴, 𝑦⟩) = 𝐴 ↔ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴))
7		vex 3353	. . 3 ⊢ 𝑥 ∈ V
8		vex 3353	. . 3 ⊢ 𝑦 ∈ V
9	7, 8	op1st 7374	. 2 ⊢ (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
10	4, 6, 9	vtocl2g 3422	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1652   ∈ wcel 2155  ⟨cop 4340  ‘cfv 6068  1^st c1st 7364

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-iota 6031  df-fun 6070  df-fv 6076  df-1st 7366

This theorem is referenced by:  ot1stg  7380  ot2ndg  7381  br1steqg  7388  1stconst  7467  mpt2sn  7470  curry2  7474  mpt2xopn0yelv  7542  mpt2xopoveq  7548  xpmapenlem  8334  1stinl  9004  1stinr  9006  fpwwe  9721  addpipq  10012  mulpipq  10015  ordpipq  10017  swrdval  13618  ruclem1  15242  qnumdenbi  15731  setsstruct  16171  oppccofval  16641  funcf2  16793  cofuval2  16812  resfval2  16818  resf1st  16819  isnat  16872  fucco  16887  homadm  16955  setcco  16998  estrcco  17035  xpcco  17089  xpchom2  17092  xpcco2  17093  evlf2  17124  curfval  17129  curf1cl  17134  uncf1  17142  uncf2  17143  diag11  17149  diag12  17150  diag2  17151  hof2fval  17161  yonedalem21  17179  yonedalem22  17184  mvmulfval  20625  imasdsf1olem  22457  ovolicc1  23574  ioombl1lem3  23618  ioombl1lem4  23619  brcgr  26071  opvtxfv  26173  fgreu  29920  fvtransport  32583  bj-elid3  33520  bj-inftyexpiinv  33529  bj-finsumval0  33581  poimirlem17  33850  poimirlem24  33857  poimirlem27  33860  rngoablo2  34130  dvhopvadd  37049  dvhopvsca  37058  dvhopaddN  37070  dvhopspN  37071  etransclem44  41132  ovnsubaddlem1  41424  ovnlecvr2  41464  ovolval5lem2  41507  rngccoALTV  42657  ringccoALTV  42720