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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 6024 | . 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 4196 | . . . . . 6 ⊢ (V ∩ dom 𝑅) = (dom 𝑅 ∩ V) | |
4 | inv1 4389 | . . . . . 6 ⊢ (dom 𝑅 ∩ V) = dom 𝑅 | |
5 | 3, 4 | eqtri 2754 | . . . . 5 ⊢ (V ∩ dom 𝑅) = dom 𝑅 |
6 | 5 | difeq1i 4113 | . . . 4 ⊢ ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋}) |
7 | 2, 6 | eqtri 2754 | . . 3 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋}) |
8 | 7 | reseq2i 5972 | . 2 ⊢ (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})) |
9 | 1, 8 | eqtr3di 2781 | 1 ⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 Vcvv 3468 ∖ cdif 3940 ∩ cin 3942 {csn 4623 dom cdm 5669 ↾ cres 5671 Rel wrel 5674 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5142 df-opab 5204 df-xp 5675 df-rel 5676 df-dm 5679 df-res 5681 |
This theorem is referenced by: funresdfunsn 7183 |
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