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| Mirrors > Home > MPE Home > Th. List > reldif | Structured version Visualization version GIF version | ||
| Description: A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.) | 
| Ref | Expression | 
|---|---|
| reldif | ⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | difss 4135 | . 2 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴 | |
| 2 | relss 5790 | . 2 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → (Rel 𝐴 → Rel (𝐴 ∖ 𝐵))) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (Rel 𝐴 → Rel (𝐴 ∖ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∖ cdif 3947 ⊆ wss 3950 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-dif 3953 df-ss 3967 df-rel 5691 | 
| This theorem is referenced by: difopab 5839 difopabOLD 5840 fundif 6614 relsdom 8993 opeldifid 32613 gsumhashmul 33065 fundmpss 35768 relbigcup 35899 vvdifopab 38262 | 
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