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Theorem br1steq 31796
 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 7232 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
41, 2, 3mp2an 708 1 (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   = wceq 1523   ∈ wcel 2030  Vcvv 3231  ⟨cop 4216   class class class wbr 4685  1st c1st 7208 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210 This theorem is referenced by:  br1steqgOLD  31798  dfdm5  31800  brtxp  32112  brpprod  32117  elfuns  32147  brimg  32169  brcup  32171  brcap  32172  brrestrict  32181
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