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Mirrors > Home > MPE Home > Th. List > reluni | Structured version Visualization version 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 4945 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
2 | 1 | releqi 5624 | . 2 ⊢ (Rel ∪ 𝐴 ↔ Rel ∪ 𝑥 ∈ 𝐴 𝑥) |
3 | reliun 5661 | . 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝑥 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) | |
4 | 2, 3 | bitri 278 | 1 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∀wral 3053 ∪ cuni 4797 ∪ ciun 4882 Rel wrel 5531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-rex 3059 df-v 3400 df-in 3851 df-ss 3861 df-uni 4798 df-iun 4884 df-rel 5533 |
This theorem is referenced by: fununi 6415 wfrrel 7990 tfrlem6 8048 bnj1379 32381 frrlem6 33446 |
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