Theorem op1st 7067

Description: Extract the first member of an ordered pair. (Contributed by NM, 5-Oct-2004.)

Hypotheses

Ref	Expression
op1st.1	⊢ 𝐴 ∈ V
op1st.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
op1st	⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴

Proof of Theorem op1st

Step	Hyp	Ref	Expression
1		1stval 7061	. 2 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = ∪ dom {⟨𝐴, 𝐵⟩}
2		op1st.1	. . 3 ⊢ 𝐴 ∈ V
3		op1st.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	op1sta 5535	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
5	1, 4	eqtri 2632	1 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴

Syntax hints:   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ⟨cop 4131  ∪ cuni 4372  dom cdm 5038  ‘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:  op1std  7069  op1stg  7071  1stval2  7076  fo1stres  7083  eloprabi  7121  algrflem  7173  xpmapenlem  8012  fseqenlem2  8731  archnq  9681  ruclem8  14805  idfu1st  16362  cofu1st  16366  xpccatid  16651  prf1st  16667  yonedalem21  16736  yonedalem22  16741  2ndcctbss  21068  upxp  21236  uptx  21238  cnheiborlem  22561  ovollb2lem  23063  ovolctb  23065  ovoliunlem2  23078  ovolshftlem1  23084  ovolscalem1  23088  ovolicc1  23091  vtxvalsnop  25716  usgraexmplvtx  25931  wlknwwlknsur  26240  wlkiswwlksur  26247  clwlkfoclwwlk  26372  ex-1st  26693  cnnvg  26917  cnnvs  26919  h2hva  27215  h2hsm  27216  hhssva  27498  hhsssm  27499  hhshsslem1  27508  eulerpartlemgvv  29765  eulerpartlemgh  29767  br1steq  30917  filnetlem3  31545  poimirlem17  32596  heiborlem8  32787  dvhvaddass  35404  dvhlveclem  35415  diblss  35477  pellexlem5  36415  pellex  36417  dvnprodlem1  38836  hoicvr  39438  hoicvrrex  39446  ovn0lem  39455  ovnhoilem1  39491  wlknwwlksnsur  41087  wlkwwlksur  41094  clwlksfoclwwlk  41270