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Theorem br1steq 35302
Description: Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
br1steq.1 𝐴 ∈ V
br1steq.2 𝐵 ∈ V
Assertion
Ref Expression
br1steq (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)

Proof of Theorem br1steq
StepHypRef Expression
1 br1steq.1 . 2 𝐴 ∈ V
2 br1steq.2 . 2 𝐵 ∈ V
3 br1steqg 8009 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
41, 2, 3mp2an 691 1 (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1534  wcel 2099  Vcvv 3469  cop 4630   class class class wbr 5142  1st c1st 7985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-1st 7987
This theorem is referenced by:  dfdm5  35304  brtxp  35412  brpprod  35417  elfuns  35447  brimg  35469  brcup  35471  brcap  35472  brrestrict  35481
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