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Theorem br1steqg 7944
Description: Uniqueness condition for the binary relation 1st. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on š¶. (Revised by Peter Mazsa, 17-Jan-2022.)
Assertion
Ref Expression
br1steqg ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (āŸØš“, šµāŸ©1st š¶ ā†” š¶ = š“))

Proof of Theorem br1steqg
StepHypRef Expression
1 op1stg 7934 . . 3 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (1st ā€˜āŸØš“, šµāŸ©) = š“)
21eqeq1d 2739 . 2 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ ((1st ā€˜āŸØš“, šµāŸ©) = š¶ ā†” š“ = š¶))
3 fo1st 7942 . . . 4 1st :Vā€“ontoā†’V
4 fofn 6759 . . . 4 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
53, 4ax-mp 5 . . 3 1st Fn V
6 opex 5422 . . 3 āŸØš“, šµāŸ© āˆˆ V
7 fnbrfvb 6896 . . 3 ((1st Fn V āˆ§ āŸØš“, šµāŸ© āˆˆ V) ā†’ ((1st ā€˜āŸØš“, šµāŸ©) = š¶ ā†” āŸØš“, šµāŸ©1st š¶))
85, 6, 7mp2an 691 . 2 ((1st ā€˜āŸØš“, šµāŸ©) = š¶ ā†” āŸØš“, šµāŸ©1st š¶)
9 eqcom 2744 . 2 (š“ = š¶ ā†” š¶ = š“)
102, 8, 93bitr3g 313 1 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (āŸØš“, šµāŸ©1st š¶ ā†” š¶ = š“))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   ā†” wb 205   āˆ§ wa 397   = wceq 1542   āˆˆ wcel 2107  Vcvv 3446  āŸØcop 4593   class class class wbr 5106   Fn wfn 6492  ā€“ontoā†’wfo 6495  ā€˜cfv 6497  1st c1st 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922
This theorem is referenced by:  br1steq  34348  fv1stcnv  34354  brxrn  36839
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