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Theorem br1steqg 7424
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 7414 . . 3 ((𝐴𝑉𝐵𝑊) → (1st ‘⟨𝐴, 𝐵⟩) = 𝐴)
21eqeq1d 2802 . 2 ((𝐴𝑉𝐵𝑊) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶𝐴 = 𝐶))
3 fo1st 7422 . . . 4 1st :V–onto→V
4 fofn 6334 . . . 4 (1st :V–onto→V → 1st Fn V)
53, 4ax-mp 5 . . 3 1st Fn V
6 opex 5124 . . 3 𝐴, 𝐵⟩ ∈ V
7 fnbrfvb 6461 . . 3 ((1st Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶))
85, 6, 7mp2an 684 . 2 ((1st ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩1st 𝐶)
9 eqcom 2807 . 2 (𝐴 = 𝐶𝐶 = 𝐴)
102, 8, 93bitr3g 305 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩1st 𝐶𝐶 = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  Vcvv 3386  cop 4375   class class class wbr 4844   Fn wfn 6097  ontowfo 6100  cfv 6102  1st c1st 7400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-fo 6108  df-fv 6110  df-1st 7402
This theorem is referenced by:  br1steq  32183  fv1stcnv  32191  brxrn  34629
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