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Theorem br1steqg 7700
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 7690 . . 3 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
21eqeq1d 2820 . 2 ((𝐴𝑉𝐵𝑊) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶𝐴 = 𝐶))
3 fo1st 7698 . . . 4 1st :V–onto→V
4 fofn 6585 . . . 4 (1st :V–onto→V → 1st Fn V)
53, 4ax-mp 5 . . 3 1st Fn V
6 opex 5347 . . 3 𝐴, 𝐵⟩ ∈ V
7 fnbrfvb 6711 . . 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 2825 . 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 1528  wcel 2105  Vcvv 3492  cop 4563   class class class wbr 5057   Fn wfn 6343  ontowfo 6346  cfv 6348  1st c1st 7676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678
This theorem is referenced by:  br1steq  32911  fv1stcnv  32917  brxrn  35506
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