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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 3512 | . . 3 ⊢ (𝐶 ∈ 𝑉 → 𝐶 ∈ V) | |
3 | elex 3512 | . . 3 ⊢ (𝐻 ∈ 𝑊 → 𝐻 ∈ V) | |
4 | simpl 485 | . . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 𝑐 = 𝐶) | |
5 | simpr 487 | . . . . . . . 8 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → ℎ = 𝐻) | |
6 | 5 | dmeqd 5774 | . . . . . . 7 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom ℎ = dom 𝐻) |
7 | 6 | dmeqd 5774 | . . . . . 6 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → dom dom ℎ = dom dom 𝐻) |
8 | 4, 7 | oveq12d 7174 | . . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → (𝑐 ↾s dom dom ℎ) = (𝐶 ↾s dom dom 𝐻)) |
9 | 5 | opeq2d 4810 | . . . . 5 ⊢ ((𝑐 = 𝐶 ∧ ℎ = 𝐻) → 〈(Hom ‘ndx), ℎ〉 = 〈(Hom ‘ndx), 𝐻〉) |
10 | 8, 9 | oveq12d 7174 | . . . 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 17081 | . . . 4 ⊢ ↾cat = (𝑐 ∈ V, ℎ ∈ V ↦ ((𝑐 ↾s dom dom ℎ) sSet 〈(Hom ‘ndx), ℎ〉)) | |
12 | ovex 7189 | . . . 4 ⊢ ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉) ∈ V | |
13 | 10, 11, 12 | ovmpoa 7305 | . . 3 ⊢ ((𝐶 ∈ V ∧ 𝐻 ∈ V) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
14 | 2, 3, 13 | syl2an 597 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ 𝑊) → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
15 | 1, 14 | syl5eq 2868 | 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 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 dom cdm 5555 ‘cfv 6355 (class class class)co 7156 ndxcnx 16480 sSet csts 16481 ↾s cress 16484 Hom chom 16576 ↾cat cresc 17078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-resc 17081 |
This theorem is referenced by: rescval2 17098 |
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