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Mirrors > Home > ILE Home > Th. List > reluni | GIF version |
Description: The union of a class is a relation iff any member is a relation. Exercise 6 of [TakeutiZaring] p. 25 and its converse. (Contributed by NM, 13-Aug-2004.) |
Ref | Expression |
---|---|
reluni | ⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniiun 3866 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
2 | 1 | releqi 4622 | . 2 ⊢ (Rel ∪ 𝐴 ↔ Rel ∪ 𝑥 ∈ 𝐴 𝑥) |
3 | reliun 4660 | . 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝑥 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) | |
4 | 2, 3 | bitri 183 | 1 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 ∀wral 2416 ∪ cuni 3736 ∪ ciun 3813 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-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 2121 |
This theorem depends on definitions: df-bi 116 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-v 2688 df-in 3077 df-ss 3084 df-uni 3737 df-iun 3815 df-rel 4546 |
This theorem is referenced by: fununi 5191 tfrlem6 6213 |
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