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Mirrors > Home > MPE Home > Th. List > relun | Structured version Visualization version GIF version |
Description: The union of two relations is a relation. Compare Exercise 5 of [TakeutiZaring] p. 25. (Contributed by NM, 12-Aug-1994.) |
Ref | Expression |
---|---|
relun | ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unss 4180 | . 2 ⊢ ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V)) | |
2 | df-rel 5679 | . . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
3 | df-rel 5679 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
4 | 2, 3 | anbi12i 627 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V))) |
5 | df-rel 5679 | . 2 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V)) | |
6 | 1, 4, 5 | 3bitr4ri 304 | 1 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 Vcvv 3470 ∪ cun 3943 ⊆ wss 3945 × cxp 5670 Rel wrel 5677 |
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 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-v 3472 df-un 3950 df-in 3952 df-ss 3962 df-rel 5679 |
This theorem is referenced by: difxp 6162 funun 6593 fununfun 6595 satfrel 34971 |
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