Theorem op1st 7839

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 7833	. 2 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = ∪ dom {⟨𝐴, 𝐵⟩}
2		op1st.1	. . 3 ⊢ 𝐴 ∈ V
3		op1st.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	op1sta 6128	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
5	1, 4	eqtri 2766	1 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴

Syntax hints:   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561  ⟨cop 4567  ∪ cuni 4839  dom cdm 5589  ‘cfv 6433  1^st c1st 7829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831

This theorem is referenced by:  op1std  7841  op1stg  7843  1stval2  7848  fo1stres  7857  opreuopreu  7876  eloprabi  7903  xpmapenlem  8931  fseqenlem2  9781  archnq  10736  ruclem8  15946  idfu1st  17594  cofu1st  17598  xpccatid  17905  prf1st  17921  yonedalem21  17991  yonedalem22  17996  2ndcctbss  22606  upxp  22774  uptx  22776  cnheiborlem  24117  ovollb2lem  24652  ovolctb  24654  ovoliunlem2  24667  ovolshftlem1  24673  ovolscalem1  24677  ovolicc1  24680  addsqnreup  26591  2sqreuop  26610  2sqreuopnn  26611  2sqreuoplt  26612  2sqreuopltb  26613  2sqreuopnnlt  26614  2sqreuopnnltb  26615  ex-1st  28808  cnnvg  29040  cnnvs  29042  h2hva  29336  h2hsm  29337  hhssva  29619  hhsssm  29620  hhshsslem1  29629  gsumhashmul  31316  eulerpartlemgvv  32343  eulerpartlemgh  32345  satfv0fvfmla0  33375  ot21std  33680  filnetlem3  34569  poimirlem17  35794  heiborlem8  35976  dvhvaddass  39111  dvhlveclem  39122  diblss  39184  pellexlem5  40655  pellex  40657  dvnprodlem1  43487  hoicvr  44086  hoicvrrex  44094  ovn0lem  44103  ovnhoilem1  44139  thincciso  46330