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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 7759 | . . . 4 β’ (π β (π Γ π) β V) |
5 | fnex 7235 | . . . 4 β’ ((π» Fn (π Γ π) β§ (π Γ π) β V) β π» β V) | |
6 | 2, 4, 5 | syl2anc 582 | . . 3 β’ (π β π» β V) |
7 | rescval.1 | . . . 4 β’ π· = (πΆ βΎcat π») | |
8 | 7 | rescval 17817 | . . 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 582 | . 2 β’ (π β π· = ((πΆ βΎs dom dom π») sSet β¨(Hom βndx), π»β©)) |
10 | 2 | fndmd 6664 | . . . . . 6 β’ (π β dom π» = (π Γ π)) |
11 | 10 | dmeqd 5912 | . . . . 5 β’ (π β dom dom π» = dom (π Γ π)) |
12 | dmxpid 5936 | . . . . 5 β’ dom (π Γ π) = π | |
13 | 11, 12 | eqtrdi 2784 | . . . 4 β’ (π β dom dom π» = π) |
14 | 13 | oveq2d 7442 | . . 3 β’ (π β (πΆ βΎs dom dom π») = (πΆ βΎs π)) |
15 | 14 | oveq1d 7441 | . 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 2768 | 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 1533 β wcel 2098 Vcvv 3473 β¨cop 4638 Γ cxp 5680 dom cdm 5682 Fn wfn 6548 βcfv 6553 (class class class)co 7426 sSet csts 17139 ndxcnx 17169 βΎs cress 17216 Hom chom 17251 βΎcat cresc 17798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-resc 17801 |
This theorem is referenced by: rescbas 17819 rescbasOLD 17820 reschom 17821 rescco 17823 resccoOLD 17824 rescabs 17825 rescabsOLD 17826 rescabs2 17827 dfrngc2 20568 dfringc2 20597 rngcresringcat 20609 rngcrescrhm 20624 rngcrescrhmALTV 47420 |
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