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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 15766 | . . . 4 ⊢ ((𝑆 ∈ 𝑊 ∧ 𝑇 ⊆ 𝑆) → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇)) | |
4 | 1, 2, 3 | syl2anc 691 | . . 3 ⊢ (𝜑 → ((𝐶 ↾s 𝑆) ↾s 𝑇) = (𝐶 ↾s 𝑇)) |
5 | 4 | oveq1d 6564 | . 2 ⊢ (𝜑 → (((𝐶 ↾s 𝑆) ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉) = ((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
6 | eqid 2610 | . . 3 ⊢ ((𝐶 ↾s 𝑆) ↾cat 𝐽) = ((𝐶 ↾s 𝑆) ↾cat 𝐽) | |
7 | ovex 6577 | . . . 4 ⊢ (𝐶 ↾s 𝑆) ∈ V | |
8 | 7 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐶 ↾s 𝑆) ∈ V) |
9 | 1, 2 | ssexd 4733 | . . 3 ⊢ (𝜑 → 𝑇 ∈ V) |
10 | rescabs2.j | . . 3 ⊢ (𝜑 → 𝐽 Fn (𝑇 × 𝑇)) | |
11 | 6, 8, 9, 10 | rescval2 16311 | . 2 ⊢ (𝜑 → ((𝐶 ↾s 𝑆) ↾cat 𝐽) = (((𝐶 ↾s 𝑆) ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
12 | eqid 2610 | . . 3 ⊢ (𝐶 ↾cat 𝐽) = (𝐶 ↾cat 𝐽) | |
13 | rescabs2.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑉) | |
14 | 12, 13, 9, 10 | rescval2 16311 | . 2 ⊢ (𝜑 → (𝐶 ↾cat 𝐽) = ((𝐶 ↾s 𝑇) sSet 〈(Hom ‘ndx), 𝐽〉)) |
15 | 5, 11, 14 | 3eqtr4d 2654 | 1 ⊢ (𝜑 → ((𝐶 ↾s 𝑆) ↾cat 𝐽) = (𝐶 ↾cat 𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 〈cop 4131 × cxp 5036 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 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-nn 10898 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-resc 16294 |
This theorem is referenced by: (None) |
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