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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 | ⊢ 𝐷 = (𝐶 ↾cat 𝐻) |
rescval2.1 | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
rescval2.2 | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
rescval2.3 | ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) |
Ref | Expression |
---|---|
rescval2 | ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescval2.1 | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
2 | rescval2.3 | . . . 4 ⊢ (𝜑 → 𝐻 Fn (𝑆 × 𝑆)) | |
3 | rescval2.2 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
4 | 3, 3 | xpexd 7240 | . . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V) |
5 | fnex 6755 | . . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) | |
6 | 2, 4, 5 | syl2anc 579 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
7 | rescval.1 | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
8 | 7 | rescval 16876 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
9 | 1, 6, 8 | syl2anc 579 | . 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
10 | fndm 6237 | . . . . . . 7 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆)) | |
11 | 2, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
12 | 11 | dmeqd 5573 | . . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
13 | dmxpid 5592 | . . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
14 | 12, 13 | syl6eq 2830 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
15 | 14 | oveq2d 6940 | . . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆)) |
16 | 15 | oveq1d 6939 | . 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉) = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
17 | 9, 16 | eqtrd 2814 | 1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 Vcvv 3398 〈cop 4404 × cxp 5355 dom cdm 5357 Fn wfn 6132 ‘cfv 6137 (class class class)co 6924 ndxcnx 16256 sSet csts 16257 ↾s cress 16260 Hom chom 16353 ↾cat cresc 16857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-resc 16860 |
This theorem is referenced by: rescbas 16878 reschom 16879 rescco 16881 rescabs 16882 rescabs2 16883 dfrngc2 42997 dfringc2 43043 rngcresringcat 43055 rngcrescrhm 43110 rngcrescrhmALTV 43128 |
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