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Mirrors > Home > MPE Home > Th. List > sectss | Structured version Visualization version GIF version |
Description: The section relation is a relation between morphisms from ๐ to ๐ and morphisms from ๐ to ๐. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
issect.b | โข ๐ต = (Baseโ๐ถ) |
issect.h | โข ๐ป = (Hom โ๐ถ) |
issect.o | โข ยท = (compโ๐ถ) |
issect.i | โข 1 = (Idโ๐ถ) |
issect.s | โข ๐ = (Sectโ๐ถ) |
issect.c | โข (๐ โ ๐ถ โ Cat) |
issect.x | โข (๐ โ ๐ โ ๐ต) |
issect.y | โข (๐ โ ๐ โ ๐ต) |
Ref | Expression |
---|---|
sectss | โข (๐ โ (๐๐๐) โ ((๐๐ป๐) ร (๐๐ป๐))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issect.b | . . 3 โข ๐ต = (Baseโ๐ถ) | |
2 | issect.h | . . 3 โข ๐ป = (Hom โ๐ถ) | |
3 | issect.o | . . 3 โข ยท = (compโ๐ถ) | |
4 | issect.i | . . 3 โข 1 = (Idโ๐ถ) | |
5 | issect.s | . . 3 โข ๐ = (Sectโ๐ถ) | |
6 | issect.c | . . 3 โข (๐ โ ๐ถ โ Cat) | |
7 | issect.x | . . 3 โข (๐ โ ๐ โ ๐ต) | |
8 | issect.y | . . 3 โข (๐ โ ๐ โ ๐ต) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | sectfval 17642 | . 2 โข (๐ โ (๐๐๐) = {โจ๐, ๐โฉ โฃ ((๐ โ (๐๐ป๐) โง ๐ โ (๐๐ป๐)) โง (๐(โจ๐, ๐โฉ ยท ๐)๐) = ( 1 โ๐))}) |
10 | opabssxp 5728 | . 2 โข {โจ๐, ๐โฉ โฃ ((๐ โ (๐๐ป๐) โง ๐ โ (๐๐ป๐)) โง (๐(โจ๐, ๐โฉ ยท ๐)๐) = ( 1 โ๐))} โ ((๐๐ป๐) ร (๐๐ป๐)) | |
11 | 9, 10 | eqsstrdi 4002 | 1 โข (๐ โ (๐๐๐) โ ((๐๐ป๐) ร (๐๐ป๐))) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 = wceq 1542 โ wcel 2107 โ wss 3914 โจcop 4596 {copab 5171 ร cxp 5635 โcfv 6500 (class class class)co 7361 Basecbs 17091 Hom chom 17152 compcco 17153 Catccat 17552 Idccid 17553 Sectcsect 17635 |
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 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-sect 17638 |
This theorem is referenced by: isinv 17651 invss 17652 oppcsect2 17670 oppcinv 17671 |
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