| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5058 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
| 2 | 1 | releqi 5787 | . 2 ⊢ (Rel ∪ 𝐴 ↔ Rel ∪ 𝑥 ∈ 𝐴 𝑥) |
| 3 | reliun 5826 | . 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 206 ∀wral 3061 ∪ cuni 4907 ∪ ciun 4991 Rel wrel 5690 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-ss 3968 df-uni 4908 df-iun 4993 df-rel 5692 |
| This theorem is referenced by: fununi 6641 frrlem6 8316 wfrrelOLD 8354 tfrlem6 8422 bnj1379 34844 |
| Copyright terms: Public domain | W3C validator |