![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4016 | . 2 ⊢ ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V)) | |
2 | df-rel 5353 | . . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
3 | df-rel 5353 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
4 | 2, 3 | anbi12i 620 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V))) |
5 | df-rel 5353 | . 2 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V)) | |
6 | 1, 4, 5 | 3bitr4ri 296 | 1 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∧ wa 386 Vcvv 3414 ∪ cun 3796 ⊆ wss 3798 × cxp 5344 Rel wrel 5351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-v 3416 df-un 3803 df-in 3805 df-ss 3812 df-rel 5353 |
This theorem is referenced by: difxp 5803 funun 6172 fununfun 6174 |
Copyright terms: Public domain | W3C validator |