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Theorem br1steq 35371
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 8019 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
41, 2, 3mp2an 690 1 (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  Vcvv 3471  cop 4636   class class class wbr 5150  1st c1st 7995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-1st 7997
This theorem is referenced by:  dfdm5  35373  brtxp  35481  brpprod  35486  elfuns  35516  brimg  35538  brcup  35540  brcap  35541  brrestrict  35550
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