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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 3490 | . . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
3 | elex 3490 | . . 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 5908 | . . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom ℎ = dom 𝐻) |
7 | 6 | dmeqd 5908 | . . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom dom ℎ = dom dom 𝐻) |
8 | 4, 7 | oveq12d 7438 | . . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑐 ↾s dom dom ℎ) = (𝐶 ↾s dom dom 𝐻)) |
9 | 5 | opeq2d 4881 | . . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 〈(Hom ‘ndx), ℎ〉 = 〈(Hom ‘ndx), 𝐻〉) |
10 | 8, 9 | oveq12d 7438 | . . . 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 17794 | . . . 4 ⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet 〈(Hom ‘ndx), ℎ〉)) | |
12 | ovex 7453 | . . . 4 ⊢ ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉) ∈ V | |
13 | 10, 11, 12 | ovmpoa 7576 | . . 3 ⊢ ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
14 | 2, 3, 13 | syl2an 595 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
15 | 1, 14 | eqtrid 2780 | 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 1534 ∈ wcel 2099 Vcvv 3471 〈cop 4635 dom cdm 5678 ‘cfv 6548 (class class class)co 7420 sSet csts 17132 ndxcnx 17162 ↾s cress 17209 Hom chom 17244 ↾cat cresc 17791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-resc 17794 |
This theorem is referenced by: rescval2 17811 |
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