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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 6048 | . 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋}))) | |
| 2 | indif1 4282 | . . . 4 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋}) | |
| 3 | incom 4209 | . . . . . 6 ⊢ (V ∩ dom 𝑅) = (dom 𝑅 ∩ V) | |
| 4 | inv1 4398 | . . . . . 6 ⊢ (dom 𝑅 ∩ V) = dom 𝑅 | |
| 5 | 3, 4 | eqtri 2765 | . . . . 5 ⊢ (V ∩ dom 𝑅) = dom 𝑅 |
| 6 | 5 | difeq1i 4122 | . . . 4 ⊢ ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋}) |
| 7 | 2, 6 | eqtri 2765 | . . 3 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋}) |
| 8 | 7 | reseq2i 5994 | . 2 ⊢ (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})) |
| 9 | 1, 8 | eqtr3di 2792 | 1 ⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 {csn 4626 dom cdm 5685 ↾ cres 5687 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-dm 5695 df-res 5697 |
| This theorem is referenced by: funresdfunsn 7209 |
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