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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 7454 | . . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V) |
5 | fnex 6957 | . . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) | |
6 | 2, 4, 5 | syl2anc 587 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
7 | rescval.1 | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
8 | 7 | rescval 17089 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
9 | 1, 6, 8 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
10 | 2 | fndmd 6427 | . . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
11 | 10 | dmeqd 5738 | . . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
12 | dmxpid 5764 | . . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
13 | 11, 12 | eqtrdi 2849 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
14 | 13 | oveq2d 7151 | . . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆)) |
15 | 14 | oveq1d 7150 | . 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉) = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
16 | 9, 15 | eqtrd 2833 | 1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 〈cop 4531 × cxp 5517 dom cdm 5519 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 ndxcnx 16472 sSet csts 16473 ↾s cress 16476 Hom chom 16568 ↾cat cresc 17070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-resc 17073 |
This theorem is referenced by: rescbas 17091 reschom 17092 rescco 17094 rescabs 17095 rescabs2 17096 dfrngc2 44596 dfringc2 44642 rngcresringcat 44654 rngcrescrhm 44709 rngcrescrhmALTV 44727 |
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