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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 4184 | . . . 4 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) | |
2 | df-res 5649 | . . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V)) | |
3 | resdm2 6187 | . . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 | |
4 | 2, 3 | eqtr3i 2763 | . . . . 5 ⊢ (𝐴 ∩ (dom 𝐴 × V)) = ◡◡𝐴 |
5 | 4 | ineq2i 4173 | . . . 4 ⊢ ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ ◡◡𝐴) |
6 | incom 4165 | . . . 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 5649 | . . . 4 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) | |
9 | dmres 5963 | . . . . . . 7 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
10 | 9 | xpeq1i 5663 | . . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V) |
11 | xpindir 5794 | . . . . . 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 4173 | . . . 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 5649 | . . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (◡◡𝐴 ∩ (𝐵 × V)) | |
16 | 7, 14, 15 | 3eqtr4i 2771 | . 2 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (◡◡𝐴 ↾ 𝐵) |
17 | rescnvcnv 6160 | . 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 3447 ∩ cin 3913 × cxp 5635 ◡ccnv 5636 dom cdm 5637 ↾ cres 5639 |
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 5260 ax-nul 5267 ax-pr 5388 |
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 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 df-dm 5647 df-rn 5648 df-res 5649 |
This theorem is referenced by: resresdm 6189 imadmres 6190 lindfres 21252 imacmp 22771 metreslem 23738 volres 24915 eccnvepres3 36796 |
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