Theorem op1st 7176

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 7170	. 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 5617	. 2 |- U. dom { <. A , B >. } = A
5	1, 4	eqtri 2644	1 |- ( 1st ` <. A , B >. ) = A

Syntax hints:   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183   U.cuni 4436   dom cdm 5114   ` cfv 5888   1stc1st 7166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168

This theorem is referenced by:  op1std  7178  op1stg  7180  1stval2  7185  fo1stres  7192  eloprabi  7232  algrflem  7286  xpmapenlem  8127  fseqenlem2  8848  archnq  9802  ruclem8  14966  idfu1st  16539  cofu1st  16543  xpccatid  16828  prf1st  16844  yonedalem21  16913  yonedalem22  16918  2ndcctbss  21258  upxp  21426  uptx  21428  cnheiborlem  22753  ovollb2lem  23256  ovolctb  23258  ovoliunlem2  23271  ovolshftlem1  23277  ovolscalem1  23281  ovolicc1  23284  wlknwwlksnsur  26776  wlkwwlksur  26783  clwlksfoclwwlk  26963  ex-1st  27301  cnnvg  27533  cnnvs  27535  h2hva  27831  h2hsm  27832  hhssva  28114  hhsssm  28115  hhshsslem1  28124  eulerpartlemgvv  30438  eulerpartlemgh  30440  br1steq  31670  filnetlem3  32375  poimirlem17  33426  heiborlem8  33617  dvhvaddass  36386  dvhlveclem  36397  diblss  36459  pellexlem5  37397  pellex  37399  dvnprodlem1  40161  hoicvr  40762  hoicvrrex  40770  ovn0lem  40779  ovnhoilem1  40815