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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 7771 | . . . 4 ⊢ (𝜑 → (𝑆 × 𝑆) ∈ V) |
| 5 | fnex 7237 | . . . 4 ⊢ ((𝐻 Fn (𝑆 × 𝑆) ∧ (𝑆 × 𝑆) ∈ V) → 𝐻 ∈ V) | |
| 6 | 2, 4, 5 | syl2anc 584 | . . 3 ⊢ (𝜑 → 𝐻 ∈ V) |
| 7 | rescval.1 | . . . 4 ⊢ 𝐷 = (𝐶 ↾cat 𝐻) | |
| 8 | 7 | rescval 17871 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐻 ∈ V) → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| 9 | 1, 6, 8 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| 10 | 2 | fndmd 6673 | . . . . . 6 ⊢ (𝜑 → dom 𝐻 = (𝑆 × 𝑆)) |
| 11 | 10 | dmeqd 5916 | . . . . 5 ⊢ (𝜑 → dom dom 𝐻 = dom (𝑆 × 𝑆)) |
| 12 | dmxpid 5941 | . . . . 5 ⊢ dom (𝑆 × 𝑆) = 𝑆 | |
| 13 | 11, 12 | eqtrdi 2793 | . . . 4 ⊢ (𝜑 → dom dom 𝐻 = 𝑆) |
| 14 | 13 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (𝐶 ↾s dom dom 𝐻) = (𝐶 ↾s 𝑆)) |
| 15 | 14 | oveq1d 7446 | . 2 ⊢ (𝜑 → ((𝐶 ↾s dom dom 𝐻) sSet 〈(Hom ‘ndx), 𝐻〉) = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| 16 | 9, 15 | eqtrd 2777 | 1 ⊢ (𝜑 → 𝐷 = ((𝐶 ↾s 𝑆) sSet 〈(Hom ‘ndx), 𝐻〉)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 × cxp 5683 dom cdm 5685 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 sSet csts 17200 ndxcnx 17230 ↾s cress 17274 Hom chom 17308 ↾cat cresc 17852 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-resc 17855 |
| This theorem is referenced by: rescbas 17873 reschom 17874 rescco 17876 rescabs 17877 rescabsOLD 17878 rescabs2 17879 dfrngc2 20628 dfringc2 20657 rngcresringcat 20669 rngcrescrhm 20684 rngcrescrhmALTV 48196 |
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