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Mirrors > Home > MPE Home > Th. List > resdmdfsn | Structured version Visualization version GIF version |
Description: Restricting a relation to its domain without a set is the same as restricting the relation to the universe without this set. (Contributed by AV, 2-Dec-2018.) |
Ref | Expression |
---|---|
resdmdfsn | ⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resindm 6029 | . 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋}))) | |
2 | indif1 4266 | . . . 4 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋}) | |
3 | incom 4195 | . . . . . 6 ⊢ (V ∩ dom 𝑅) = (dom 𝑅 ∩ V) | |
4 | inv1 4390 | . . . . . 6 ⊢ (dom 𝑅 ∩ V) = dom 𝑅 | |
5 | 3, 4 | eqtri 2753 | . . . . 5 ⊢ (V ∩ dom 𝑅) = dom 𝑅 |
6 | 5 | difeq1i 4110 | . . . 4 ⊢ ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋}) |
7 | 2, 6 | eqtri 2753 | . . 3 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋}) |
8 | 7 | reseq2i 5976 | . 2 ⊢ (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})) |
9 | 1, 8 | eqtr3di 2780 | 1 ⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3463 ∖ cdif 3936 ∩ cin 3938 {csn 4624 dom cdm 5672 ↾ cres 5674 Rel wrel 5677 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5144 df-opab 5206 df-xp 5678 df-rel 5679 df-dm 5682 df-res 5684 |
This theorem is referenced by: funresdfunsn 7194 |
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