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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 5023 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
2 | 1 | releqi 5738 | . 2 ⊢ (Rel ∪ 𝐴 ↔ Rel ∪ 𝑥 ∈ 𝐴 𝑥) |
3 | reliun 5777 | . 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝑥 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) | |
4 | 2, 3 | bitri 275 | 1 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wral 3065 ∪ cuni 4870 ∪ ciun 4959 Rel wrel 5643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-v 3450 df-in 3922 df-ss 3932 df-uni 4871 df-iun 4961 df-rel 5645 |
This theorem is referenced by: fununi 6581 frrlem6 8227 wfrrelOLD 8265 tfrlem6 8333 bnj1379 33482 |
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