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 4573 | . . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥 | |
2 | 1 | releqi 5202 | . 2 ⊢ (Rel ∪ 𝐴 ↔ Rel ∪ 𝑥 ∈ 𝐴 𝑥) |
3 | reliun 5239 | . 2 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 𝑥 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) | |
4 | 2, 3 | bitri 264 | 1 ⊢ (Rel ∪ 𝐴 ↔ ∀𝑥 ∈ 𝐴 Rel 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wral 2912 ∪ cuni 4436 ∪ ciun 4520 Rel wrel 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-v 3202 df-in 3581 df-ss 3588 df-uni 4437 df-iun 4522 df-rel 5121 |
This theorem is referenced by: fununi 5964 wfrrel 7420 tfrlem6 7478 bnj1379 30901 frrlem5b 31785 frrlem6 31789 |
Copyright terms: Public domain | W3C validator |