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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 3786 | . . . 4 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) | |
2 | df-res 5050 | . . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = (𝐴 ∩ (dom 𝐴 × V)) | |
3 | resdm2 5542 | . . . . . 6 ⊢ (𝐴 ↾ dom 𝐴) = ◡◡𝐴 | |
4 | 2, 3 | eqtr3i 2634 | . . . . 5 ⊢ (𝐴 ∩ (dom 𝐴 × V)) = ◡◡𝐴 |
5 | 4 | ineq2i 3773 | . . . 4 ⊢ ((𝐵 × V) ∩ (𝐴 ∩ (dom 𝐴 × V))) = ((𝐵 × V) ∩ ◡◡𝐴) |
6 | incom 3767 | . . . 4 ⊢ ((𝐵 × V) ∩ ◡◡𝐴) = (◡◡𝐴 ∩ (𝐵 × V)) | |
7 | 1, 5, 6 | 3eqtri 2636 | . . 3 ⊢ (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) = (◡◡𝐴 ∩ (𝐵 × V)) |
8 | df-res 5050 | . . . 4 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) | |
9 | dmres 5339 | . . . . . . 7 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴) | |
10 | 9 | xpeq1i 5059 | . . . . . 6 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 ∩ dom 𝐴) × V) |
11 | xpindir 5178 | . . . . . 6 ⊢ ((𝐵 ∩ dom 𝐴) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V)) | |
12 | 10, 11 | eqtri 2632 | . . . . 5 ⊢ (dom (𝐴 ↾ 𝐵) × V) = ((𝐵 × V) ∩ (dom 𝐴 × V)) |
13 | 12 | ineq2i 3773 | . . . 4 ⊢ (𝐴 ∩ (dom (𝐴 ↾ 𝐵) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) |
14 | 8, 13 | eqtri 2632 | . . 3 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ∩ ((𝐵 × V) ∩ (dom 𝐴 × V))) |
15 | df-res 5050 | . . 3 ⊢ (◡◡𝐴 ↾ 𝐵) = (◡◡𝐴 ∩ (𝐵 × V)) | |
16 | 7, 14, 15 | 3eqtr4i 2642 | . 2 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (◡◡𝐴 ↾ 𝐵) |
17 | rescnvcnv 5515 | . 2 ⊢ (◡◡𝐴 ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
18 | 16, 17 | eqtri 2632 | 1 ⊢ (𝐴 ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 Vcvv 3173 ∩ cin 3539 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ↾ cres 5040 |
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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 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-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 |
This theorem is referenced by: imadmres 5544 lindfres 19981 imacmp 21010 metreslem 21977 volres 23103 uhgrares 25837 umgrares 25853 usgrares 25898 resresdm 40319 |
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