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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 6031 | . 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋}))) | |
2 | indif1 4270 | . . . 4 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋}) | |
3 | incom 4199 | . . . . . 6 ⊢ (V ∩ dom 𝑅) = (dom 𝑅 ∩ V) | |
4 | inv1 4392 | . . . . . 6 ⊢ (dom 𝑅 ∩ V) = dom 𝑅 | |
5 | 3, 4 | eqtri 2754 | . . . . 5 ⊢ (V ∩ dom 𝑅) = dom 𝑅 |
6 | 5 | difeq1i 4114 | . . . 4 ⊢ ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋}) |
7 | 2, 6 | eqtri 2754 | . . 3 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋}) |
8 | 7 | reseq2i 5978 | . 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 1534 Vcvv 3462 ∖ cdif 3943 ∩ cin 3945 {csn 4623 dom cdm 5674 ↾ cres 5676 Rel wrel 5679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-br 5146 df-opab 5208 df-xp 5680 df-rel 5681 df-dm 5684 df-res 5686 |
This theorem is referenced by: funresdfunsn 7195 |
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