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Mirrors > Home > ILE Home > Th. List > relun | 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 3245 | . 2 ⊢ ((𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V)) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V)) | |
2 | df-rel 4541 | . . 3 ⊢ (Rel 𝐴 ↔ 𝐴 ⊆ (V × V)) | |
3 | df-rel 4541 | . . 3 ⊢ (Rel 𝐵 ↔ 𝐵 ⊆ (V × V)) | |
4 | 2, 3 | anbi12i 455 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) ↔ (𝐴 ⊆ (V × V) ∧ 𝐵 ⊆ (V × V))) |
5 | df-rel 4541 | . 2 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (𝐴 ∪ 𝐵) ⊆ (V × V)) | |
6 | 1, 4, 5 | 3bitr4ri 212 | 1 ⊢ (Rel (𝐴 ∪ 𝐵) ↔ (Rel 𝐴 ∧ Rel 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 Vcvv 2681 ∪ cun 3064 ⊆ wss 3066 × cxp 4532 Rel wrel 4539 |
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-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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-rel 4541 |
This theorem is referenced by: funun 5162 |
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