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| Mirrors > Home > MPE Home > Th. List > rescval | 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) |
| Ref | Expression |
|---|---|
| rescval | ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rescval.1 | . 2 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
| 2 | elex 3478 | . . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
| 3 | elex 3478 | . . 3 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V) | |
| 4 | simpl 482 | . . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 𝑐 = 𝐶) | |
| 5 | simpr 484 | . . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻) | |
| 6 | 5 | dmeqd 5882 | . . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom ℎ = dom 𝐻) |
| 7 | 6 | dmeqd 5882 | . . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom dom ℎ = dom dom 𝐻) |
| 8 | 4, 7 | oveq12d 7417 | . . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑐 ↾s dom dom ℎ) = (𝐶 ↾s dom dom 𝐻)) |
| 9 | 5 | opeq2d 4853 | . . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 〈(Hom ‘ndx), ℎ〉 = 〈(Hom ‘ndx), 𝐻〉) |
| 10 | 8, 9 | oveq12d 7417 | . . . 4 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ((𝑐 ↾s dom dom ℎ) sSet 〈(Hom ‘ndx), ℎ〉) = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| 11 | df-resc 17809 | . . . 4 ⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet 〈(Hom ‘ndx), ℎ〉)) | |
| 12 | ovex 7432 | . . . 4 ⊢ ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉) ∈ V | |
| 13 | 10, 11, 12 | ovmpoa 7556 | . . 3 ⊢ ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| 14 | 2, 3, 13 | syl2an 596 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| 15 | 1, 14 | eqtrid 2781 | 1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 Vcvv 3457 〈cop 4605 dom cdm 5651 ‘cfv 6527 (class class class)co 7399 sSet csts 17167 ndxcnx 17197 ↾s cress 17236 Hom chom 17267 ↾cat cresc 17806 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5263 ax-nul 5273 ax-pr 5399 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3414 df-v 3459 df-sbc 3764 df-dif 3927 df-un 3929 df-ss 3941 df-nul 4307 df-if 4499 df-sn 4600 df-pr 4602 df-op 4606 df-uni 4881 df-br 5117 df-opab 5179 df-id 5545 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-iota 6480 df-fun 6529 df-fv 6535 df-ov 7402 df-oprab 7403 df-mpo 7404 df-resc 17809 |
| This theorem is referenced by: rescval2 17826 |
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