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Mirrors > Home > MPE Home > Th. List > rescval2 | Structured version Visualization version GIF version |
Description: Value of the category restriction. (Contributed by Mario Carneiro, 4-Jan-2017.) |
Ref | Expression |
---|---|
rescval.1 | β’ π· = (πΆ βΎcat π») |
rescval2.1 | β’ (π β πΆ β π) |
rescval2.2 | β’ (π β π β π) |
rescval2.3 | β’ (π β π» Fn (π Γ π)) |
Ref | Expression |
---|---|
rescval2 | β’ (π β π· = ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescval2.1 | . . 3 β’ (π β πΆ β π) | |
2 | rescval2.3 | . . . 4 β’ (π β π» Fn (π Γ π)) | |
3 | rescval2.2 | . . . . 5 β’ (π β π β π) | |
4 | 3, 3 | xpexd 7689 | . . . 4 β’ (π β (π Γ π) β V) |
5 | fnex 7171 | . . . 4 β’ ((π» Fn (π Γ π) β§ (π Γ π) β V) β π» β V) | |
6 | 2, 4, 5 | syl2anc 585 | . . 3 β’ (π β π» β V) |
7 | rescval.1 | . . . 4 β’ π· = (πΆ βΎcat π») | |
8 | 7 | rescval 17718 | . . 3 β’ ((πΆ β π β§ π» β V) β π· = ((πΆ βΎs dom dom π») sSet β¨(Hom βndx), π»β©)) |
9 | 1, 6, 8 | syl2anc 585 | . 2 β’ (π β π· = ((πΆ βΎs dom dom π») sSet β¨(Hom βndx), π»β©)) |
10 | 2 | fndmd 6611 | . . . . . 6 β’ (π β dom π» = (π Γ π)) |
11 | 10 | dmeqd 5865 | . . . . 5 β’ (π β dom dom π» = dom (π Γ π)) |
12 | dmxpid 5889 | . . . . 5 β’ dom (π Γ π) = π | |
13 | 11, 12 | eqtrdi 2789 | . . . 4 β’ (π β dom dom π» = π) |
14 | 13 | oveq2d 7377 | . . 3 β’ (π β (πΆ βΎs dom dom π») = (πΆ βΎs π)) |
15 | 14 | oveq1d 7376 | . 2 β’ (π β ((πΆ βΎs dom dom π») sSet β¨(Hom βndx), π»β©) = ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©)) |
16 | 9, 15 | eqtrd 2773 | 1 β’ (π β π· = ((πΆ βΎs π) sSet β¨(Hom βndx), π»β©)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3447 β¨cop 4596 Γ cxp 5635 dom cdm 5637 Fn wfn 6495 βcfv 6500 (class class class)co 7361 sSet csts 17043 ndxcnx 17073 βΎs cress 17120 Hom chom 17152 βΎcat cresc 17699 |
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-resc 17702 |
This theorem is referenced by: rescbas 17720 rescbasOLD 17721 reschom 17722 rescco 17724 resccoOLD 17725 rescabs 17726 rescabsOLD 17727 rescabs2 17728 dfrngc2 46360 dfringc2 46406 rngcresringcat 46418 rngcrescrhm 46473 rngcrescrhmALTV 46491 |
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