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Theorem br2ndeqg 7914
Description: Uniqueness condition for the binary relation 2nd. (Contributed by Scott Fenton, 2-Jul-2020.) Revised to remove sethood hypothesis on š¶. (Revised by Peter Mazsa, 17-Jan-2022.)
Assertion
Ref Expression
br2ndeqg ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (āŸØš“, šµāŸ©2nd š¶ ā†” š¶ = šµ))

Proof of Theorem br2ndeqg
StepHypRef Expression
1 op2ndg 7904 . . 3 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (2nd ā€˜āŸØš“, šµāŸ©) = šµ)
21eqeq1d 2738 . 2 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ ((2nd ā€˜āŸØš“, šµāŸ©) = š¶ ā†” šµ = š¶))
3 fo2nd 7912 . . . 4 2nd :Vā€“ontoā†’V
4 fofn 6735 . . . 4 (2nd :Vā€“ontoā†’V ā†’ 2nd Fn V)
53, 4ax-mp 5 . . 3 2nd Fn V
6 opex 5403 . . 3 āŸØš“, šµāŸ© āˆˆ V
7 fnbrfvb 6872 . . 3 ((2nd Fn V āˆ§ āŸØš“, šµāŸ© āˆˆ V) ā†’ ((2nd ā€˜āŸØš“, šµāŸ©) = š¶ ā†” āŸØš“, šµāŸ©2nd š¶))
85, 6, 7mp2an 689 . 2 ((2nd ā€˜āŸØš“, šµāŸ©) = š¶ ā†” āŸØš“, šµāŸ©2nd š¶)
9 eqcom 2743 . 2 (šµ = š¶ ā†” š¶ = šµ)
102, 8, 93bitr3g 312 1 ((š“ āˆˆ š‘‰ āˆ§ šµ āˆˆ š‘Š) ā†’ (āŸØš“, šµāŸ©2nd š¶ ā†” š¶ = šµ))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   ā†” wb 205   āˆ§ wa 396   = wceq 1540   āˆˆ wcel 2105  Vcvv 3441  āŸØcop 4578   class class class wbr 5089   Fn wfn 6468  ā€“ontoā†’wfo 6471  ā€˜cfv 6473  2nd c2nd 7890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369  ax-un 7642
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-fo 6479  df-fv 6481  df-2nd 7892
This theorem is referenced by:  br2ndeq  33973  fv2ndcnv  33979  brxrn  36634
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