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Mirrors > Home > MPE Home > Th. List > br1steqg | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
br1steqg | ā¢ ((š“ ā š ā§ šµ ā š) ā (āØš“, šµā©1st š¶ ā š¶ = š“)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | op1stg 7934 | . . 3 ā¢ ((š“ ā š ā§ šµ ā š) ā (1st āāØš“, šµā©) = š“) | |
2 | 1 | eqeq1d 2739 | . 2 ā¢ ((š“ ā š ā§ šµ ā š) ā ((1st āāØš“, šµā©) = š¶ ā š“ = š¶)) |
3 | fo1st 7942 | . . . 4 ā¢ 1st :VāontoāV | |
4 | fofn 6759 | . . . 4 ā¢ (1st :VāontoāV ā 1st Fn V) | |
5 | 3, 4 | ax-mp 5 | . . 3 ā¢ 1st Fn V |
6 | opex 5422 | . . 3 ā¢ āØš“, šµā© ā V | |
7 | fnbrfvb 6896 | . . 3 ā¢ ((1st Fn V ā§ āØš“, šµā© ā V) ā ((1st āāØš“, šµā©) = š¶ ā āØš“, šµā©1st š¶)) | |
8 | 5, 6, 7 | mp2an 691 | . 2 ā¢ ((1st āāØš“, šµā©) = š¶ ā āØš“, šµā©1st š¶) |
9 | eqcom 2744 | . 2 ā¢ (š“ = š¶ ā š¶ = š“) | |
10 | 2, 8, 9 | 3bitr3g 313 | 1 ā¢ ((š“ ā š ā§ šµ ā š) ā (āØš“, šµā©1st š¶ ā š¶ = š“)) |
Colors of variables: wff setvar class |
Syntax hints: ā wi 4 ā wb 205 ā§ wa 397 = wceq 1542 ā wcel 2107 Vcvv 3446 āØcop 4593 class class class wbr 5106 Fn wfn 6492 āontoāwfo 6495 ācfv 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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