Theorem op1st 7679

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 7673	. 2 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = ∪ dom {⟨𝐴, 𝐵⟩}
2		op1st.1	. . 3 ⊢ 𝐴 ∈ V
3		op1st.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	op1sta 6049	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴
5	1, 4	eqtri 2821	1 ⊢ (1st ‘⟨𝐴, 𝐵⟩) = 𝐴

Syntax hints:   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531  ∪ cuni 4800  dom cdm 5519  ‘cfv 6324  1^st c1st 7669

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-1st 7671

This theorem is referenced by:  op1std  7681  op1stg  7683  1stval2  7688  fo1stres  7697  opreuopreu  7716  eloprabi  7743  algrflem  7802  xpmapenlem  8668  fseqenlem2  9436  archnq  10391  ruclem8  15582  idfu1st  17141  cofu1st  17145  xpccatid  17430  prf1st  17446  yonedalem21  17515  yonedalem22  17520  2ndcctbss  22060  upxp  22228  uptx  22230  cnheiborlem  23559  ovollb2lem  24092  ovolctb  24094  ovoliunlem2  24107  ovolshftlem1  24113  ovolscalem1  24117  ovolicc1  24120  addsqnreup  26027  2sqreuop  26046  2sqreuopnn  26047  2sqreuoplt  26048  2sqreuopltb  26049  2sqreuopnnlt  26050  2sqreuopnnltb  26051  ex-1st  28229  cnnvg  28461  cnnvs  28463  h2hva  28757  h2hsm  28758  hhssva  29040  hhsssm  29041  hhshsslem1  29050  gsumhashmul  30741  eulerpartlemgvv  31744  eulerpartlemgh  31746  satfv0fvfmla0  32773  filnetlem3  33841  poimirlem17  35074  heiborlem8  35256  dvhvaddass  38393  dvhlveclem  38404  diblss  38466  pellexlem5  39774  pellex  39776  dvnprodlem1  42588  hoicvr  43187  hoicvrrex  43195  ovn0lem  43204  ovnhoilem1  43240