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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 | xpexg 6858 | . . . . 5 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑆 ∈ 𝑊) → (𝑆 × 𝑆) ∈ V) | |
5 | 3, 3, 4 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V) |
6 | fnex 6386 | . . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) | |
7 | 2, 5, 6 | syl2anc 691 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
8 | rescval.1 | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
9 | 8 | rescval 16310 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
10 | 1, 7, 9 | syl2anc 691 | . 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
11 | fndm 5904 | . . . . . . 7 ⊢ (𝐻 Fn (𝑆 × 𝑆) → dom 𝐻 = (𝑆 × 𝑆)) | |
12 | 2, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
13 | 12 | dmeqd 5248 | . . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
14 | dmxpid 5266 | . . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
15 | 13, 14 | syl6eq 2660 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
16 | 15 | oveq2d 6565 | . . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆)) |
17 | 16 | oveq1d 6564 | . 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉) = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
18 | 10, 17 | eqtrd 2644 | 1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 × cxp 5036 dom cdm 5038 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 ndxcnx 15692 sSet csts 15693 ↾s cress 15696 Hom chom 15779 ↾cat cresc 16291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-resc 16294 |
This theorem is referenced by: rescbas 16312 reschom 16313 rescco 16315 rescabs 16316 rescabs2 16317 dfrngc2 41764 dfringc2 41810 rngcresringcat 41822 rngcrescrhm 41877 rngcrescrhmALTV 41896 |
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