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Mirrors > Home > MPE Home > Th. List > rescabs2 | Structured version Visualization version GIF version |
Description: Restriction absorption law. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
rescabs2.c | ⊢ (𝜑 → 𝐶 ∈ 𝑉) |
rescabs2.j | ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) |
rescabs2.s | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
rescabs2.t | ⊢ (𝜑 → 𝑇 ⊆ 𝑆) |
Ref | Expression |
---|---|
rescabs2 | ⊢ (𝜑 → ((𝐶 ↾s 𝑆) ↾cat 𝐽) = (𝐶 ↾cat 𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rescabs2.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
2 | rescabs2.t | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑆) | |
3 | ressabs 17084 | . . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑇 ⊆ 𝑆) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇)) | |
4 | 1, 2, 3 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇)) |
5 | 4 | oveq1d 7366 | . 2 ⊢ (𝜑 → (((𝐶 ↾s 𝑆) ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉) = ((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
6 | eqid 2736 | . . 3 ⊢ ((𝐶 ↾s 𝑆) ↾cat 𝐽) = ((𝐶 ↾s 𝑆) ↾cat 𝐽) | |
7 | ovexd 7386 | . . 3 ⊢ (𝜑 → (𝐶 ↾s 𝑆) ∈ V) | |
8 | 1, 2 | ssexd 5279 | . . 3 ⊢ (𝜑 → 𝑇 ∈ V) |
9 | rescabs2.j | . . 3 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) | |
10 | 6, 7, 8, 9 | rescval2 17665 | . 2 ⊢ (𝜑 → ((𝐶 ↾s 𝑆) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
11 | eqid 2736 | . . 3 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽) | |
12 | rescabs2.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
13 | 11, 12, 8, 9 | rescval2 17665 | . 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
14 | 5, 10, 13 | 3eqtr4d 2786 | 1 ⊢ (𝜑 → ((𝐶 ↾s 𝑆) ↾cat 𝐽) = (𝐶 ↾cat 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ⊆ wss 3908 〈cop 4590 × cxp 5629 Fn wfn 6488 ‘cfv 6493 (class class class)co 7351 sSet csts 16989 ndxcnx 17019 ↾s cress 17066 Hom chom 17098 ↾cat cresc 17645 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-1cn 11067 ax-addcl 11069 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-nn 12112 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-resc 17648 |
This theorem is referenced by: (None) |
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