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Mirrors > Home > MPE Home > Th. List > resdmres | Structured version Visualization version GIF version |
Description: Restriction to the domain of a restriction. (Contributed by NM, 8-Apr-2007.) |
Ref | Expression |
---|---|
resdmres | ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | in12 4221 | . . . 4 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) | |
2 | df-res 5689 | . . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V)) | |
3 | resdm2 6231 | . . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 | |
4 | 2, 3 | eqtr3i 2763 | . . . . 5 ⊢ (𝐴 ∩ (dom 𝐴 × V)) = ◡◡𝐴 |
5 | 4 | ineq2i 4210 | . . . 4 ⊢ ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ ◡◡𝐴) |
6 | incom 4202 | . . . 4 ⊢ ((𝐵 × V) ∩ ◡◡𝐴) = (◡◡𝐴 ∩ (𝐵 × V)) | |
7 | 1, 5, 6 | 3eqtri 2765 | . . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (◡◡𝐴 ∩ (𝐵 × V)) |
8 | df-res 5689 | . . . 4 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) | |
9 | dmres 6004 | . . . . . . 7 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
10 | 9 | xpeq1i 5703 | . . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V) |
11 | xpindir 5835 | . . . . . 6 ⊢ ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V)) | |
12 | 10, 11 | eqtri 2761 | . . . . 5 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V)) |
13 | 12 | ineq2i 4210 | . . . 4 ⊢ (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) |
14 | 8, 13 | eqtri 2761 | . . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) |
15 | df-res 5689 | . . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (◡◡𝐴 ∩ (𝐵 × V)) | |
16 | 7, 14, 15 | 3eqtr4i 2771 | . 2 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (◡◡𝐴 ↾ 𝐵) |
17 | rescnvcnv 6204 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
18 | 16, 17 | eqtri 2761 | 1 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Vcvv 3475 ∩ cin 3948 × cxp 5675 ◡ccnv 5676 dom cdm 5677 ↾ cres 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 df-rn 5688 df-res 5689 |
This theorem is referenced by: resresdm 6233 imadmres 6234 lindfres 21378 imacmp 22901 metreslem 23868 volres 25045 eccnvepres3 37154 |
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