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Mirrors > Home > ILE Home > Th. List > resdmdfsn | 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 | indif1 3321 | . . . 4 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋}) | |
2 | incom 3268 | . . . . . 6 ⊢ (V ∩ dom 𝑅) = (dom 𝑅 ∩ V) | |
3 | inv1 3399 | . . . . . 6 ⊢ (dom 𝑅 ∩ V) = dom 𝑅 | |
4 | 2, 3 | eqtri 2160 | . . . . 5 ⊢ (V ∩ dom 𝑅) = dom 𝑅 |
5 | 4 | difeq1i 3190 | . . . 4 ⊢ ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋}) |
6 | 1, 5 | eqtri 2160 | . . 3 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋}) |
7 | 6 | reseq2i 4816 | . 2 ⊢ (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})) |
8 | resindm 4861 | . 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋}))) | |
9 | 7, 8 | syl5reqr 2187 | 1 ⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Vcvv 2686 ∖ cdif 3068 ∩ cin 3070 {csn 3527 dom cdm 4539 ↾ cres 4541 Rel wrel 4544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-xp 4545 df-rel 4546 df-dm 4549 df-res 4551 |
This theorem is referenced by: funresdfunsnss 5623 funresdfunsndc 6402 |
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