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Theorem op1st 5692
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 op1st.1 . . . 4 A V
2 op1st.2 . . . 4 B V
3 opexg 3934 . . . 4 ((A V B V) → ⟨A, B V)
41, 2, 3mp2an 404 . . 3 A, B V
5 1stvalg 5688 . . 3 (⟨A, B V → (1st ‘⟨A, B⟩) = dom {⟨A, B⟩})
64, 5ax-mp 7 . 2 (1st ‘⟨A, B⟩) = dom {⟨A, B⟩}
71, 2op1sta 4725 . 2 dom {⟨A, B⟩} = A
86, 7eqtri 2038 1 (1st ‘⟨A, B⟩) = A
Colors of variables: wff set class
Syntax hints:   = wceq 1226   wcel 1370  Vcvv 2531  {csn 3346  cop 3349   cuni 3550  dom cdm 4268  cfv 4825  1st c1st 5684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fv 4833  df-1st 5686
This theorem is referenced by:  op1std  5694  op1stg  5696  1stval2  5701  fo1stresm  5707  eloprabi  5741  algrflem  5769  genpelvl  6360  1prl  6399  ltexprlemell  6429  recexprlemell  6450
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