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Theorem br1steqg 7702
 Description: Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on 𝐶. (Revised by Peter Mazsa, 17-Jan-2022.)
Assertion
Ref Expression
br1steqg ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))

Proof of Theorem br1steqg
StepHypRef Expression
1 op1stg 7692 . . 3 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
21eqeq1d 2828 . 2 ((𝐴𝑉𝐵𝑊) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶𝐴 = 𝐶))
3 fo1st 7700 . . . 4 1st :V–onto→V
4 fofn 6589 . . . 4 (1st :V–onto→V → 1st Fn V)
53, 4ax-mp 5 . . 3 1st Fn V
6 opex 5353 . . 3 𝐴, 𝐵⟩ ∈ V
7 fnbrfvb 6715 . . 3 ((1st Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶))
85, 6, 7mp2an 688 . 2 ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶)
9 eqcom 2833 . 2 (𝐴 = 𝐶𝐶 = 𝐴)
102, 8, 93bitr3g 314 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1530   ∈ wcel 2107  Vcvv 3500  ⟨cop 4570   class class class wbr 5063   Fn wfn 6347  –onto→wfo 6350  ‘cfv 6352  1st c1st 7678 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fo 6358  df-fv 6360  df-1st 7680 This theorem is referenced by:  br1steq  32901  fv1stcnv  32907  brxrn  35496
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