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Mirrors > Home > MPE Home > Th. List > br2ndeqg | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
br2ndeqg | ā¢ ((š“ ā š ā§ šµ ā š) ā (āØš“, šµā©2nd š¶ ā š¶ = šµ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op2ndg 7904 | . . 3 ā¢ ((š“ ā š ā§ šµ ā š) ā (2nd āāØš“, šµā©) = šµ) | |
2 | 1 | eqeq1d 2738 | . 2 ā¢ ((š“ ā š ā§ šµ ā š) ā ((2nd āāØš“, šµā©) = š¶ ā šµ = š¶)) |
3 | fo2nd 7912 | . . . 4 ā¢ 2nd :VāontoāV | |
4 | fofn 6735 | . . . 4 ā¢ (2nd :VāontoāV ā 2nd Fn V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ā¢ 2nd Fn V |
6 | opex 5403 | . . 3 ā¢ āØš“, šµā© ā V | |
7 | fnbrfvb 6872 | . . 3 ā¢ ((2nd Fn V ā§ āØš“, šµā© ā V) ā ((2nd āāØš“, šµā©) = š¶ ā āØš“, šµā©2nd š¶)) | |
8 | 5, 6, 7 | mp2an 689 | . 2 ā¢ ((2nd āāØš“, šµā©) = š¶ ā āØš“, šµā©2nd š¶) |
9 | eqcom 2743 | . 2 ā¢ (šµ = š¶ ā š¶ = šµ) | |
10 | 2, 8, 9 | 3bitr3g 312 | 1 ā¢ ((š“ ā š ā§ šµ ā š) ā (āØš“, šµā©2nd š¶ ā š¶ = šµ)) |
Colors of variables: wff setvar class |
Syntax hints: ā wi 4 ā wb 205 ā§ wa 396 = wceq 1540 ā wcel 2105 Vcvv 3441 āØcop 4578 class class class wbr 5089 Fn wfn 6468 āontoāwfo 6471 ācfv 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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