Theorem op1st 6807

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

Hypotheses

Ref	Expression
op1st.1	|- A e. _V
op1st.2	|- B e. _V

Assertion

Ref	Expression
op1st	|- ( 1st ` <. A , B >. ) = A

Proof of Theorem op1st

Step	Hyp	Ref	Expression
1		1stval 6801	. 2 |- ( 1st ` <. A , B >. ) = U. dom { <. A , B >. }
2		op1st.1	. . 3 |- A e. _V
3		op1st.2	. . 3 |- B e. _V
4	2, 3	op1sta 5496	. 2 |- U. dom { <. A , B >. } = A
5	1, 4	eqtri 2486	1 |- ( 1st ` <. A , B >. ) = A

Syntax hints:   = wceq 1395   e. wcel 1819   _Vcvv 3109   {csn 4032   <.cop 4038   U.cuni 4251   dom cdm 5008   ` cfv 5594   1stc1st 6797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fv 5602  df-1st 6799

This theorem is referenced by:  op1std  6809  op1stg  6811  1stval2  6816  fo1stres  6823  eloprabi  6861  algrflem  6908  xpmapenlem  7703  fseqenlem2  8423  archnq  9375  ruclem8  13982  idfu1st  15295  cofu1st  15299  xpccatid  15584  prf1st  15600  yonedalem21  15669  yonedalem22  15674  2ndcctbss  20082  upxp  20250  uptx  20252  cnheiborlem  21580  ovollb2lem  22025  ovolctb  22027  ovoliunlem2  22040  ovolshftlem1  22046  ovolscalem1  22050  ovolicc1  22053  usgraexmplvtx  24529  wlknwwlknsur  24839  wlkiswwlksur  24846  clwlkfoclwwlk  24972  ex-1st  25292  cnnvg  25710  cnnvs  25713  h2hva  26018  h2hsm  26019  hhssva  26302  hhsssm  26303  hhshsslem1  26310  eulerpartlemgvv  28512  eulerpartlemgh  28514  br1steq  29421  filnetlem3  30403  heiborlem8  30519  pellexlem5  30973  pellex  30975  dvnprodlem1  31946  usgfis  32708  usgfisALT  32712  dvhvaddass  36967  dvhlveclem  36978  diblss  37040