MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  op1st Structured version   Visualization version   GIF version

Theorem op1st 7689
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
StepHypRef Expression
1 1stval 7683 . 2 (1st ‘⟨𝐴, 𝐵⟩) = dom {⟨𝐴, 𝐵⟩}
2 op1st.1 . . 3 𝐴 ∈ V
3 op1st.2 . . 3 𝐵 ∈ V
42, 3op1sta 6075 . 2 dom {⟨𝐴, 𝐵⟩} = 𝐴
51, 4eqtri 2842 1 (1st ‘⟨𝐴, 𝐵⟩) = 𝐴
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  Vcvv 3493  {csn 4559  cop 4565   cuni 4830  dom cdm 5548  cfv 6348  1st c1st 7679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-1st 7681
This theorem is referenced by:  op1std  7691  op1stg  7693  1stval2  7698  fo1stres  7707  opreuopreu  7726  eloprabi  7753  algrflem  7811  xpmapenlem  8676  fseqenlem2  9443  archnq  10394  ruclem8  15582  idfu1st  17141  cofu1st  17145  xpccatid  17430  prf1st  17446  yonedalem21  17515  yonedalem22  17520  2ndcctbss  22055  upxp  22223  uptx  22225  cnheiborlem  23550  ovollb2lem  24081  ovolctb  24083  ovoliunlem2  24096  ovolshftlem1  24102  ovolscalem1  24106  ovolicc1  24109  addsqnreup  26011  2sqreuop  26030  2sqreuopnn  26031  2sqreuoplt  26032  2sqreuopltb  26033  2sqreuopnnlt  26034  2sqreuopnnltb  26035  ex-1st  28215  cnnvg  28447  cnnvs  28449  h2hva  28743  h2hsm  28744  hhssva  29026  hhsssm  29027  hhshsslem1  29036  eulerpartlemgvv  31622  eulerpartlemgh  31624  satfv0fvfmla0  32648  filnetlem3  33716  poimirlem17  34896  heiborlem8  35083  dvhvaddass  38220  dvhlveclem  38231  diblss  38293  pellexlem5  39415  pellex  39417  dvnprodlem1  42215  hoicvr  42815  hoicvrrex  42823  ovn0lem  42832  ovnhoilem1  42868
  Copyright terms: Public domain W3C validator