![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > relcnvfld | GIF version |
Description: if 𝑅 is a relation, its double union equals the double union of its converse. (Contributed by FL, 5-Jan-2009.) |
Ref | Expression |
---|---|
relcnvfld | ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relfld 5025 | . 2 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = (dom 𝑅 ∪ ran 𝑅)) | |
2 | unidmrn 5029 | . 2 ⊢ ∪ ∪ ◡𝑅 = (dom 𝑅 ∪ ran 𝑅) | |
3 | 1, 2 | syl6eqr 2165 | 1 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1314 ∪ cun 3035 ∪ cuni 3702 ◡ccnv 4498 dom cdm 4499 ran crn 4500 Rel wrel 4504 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 |
This theorem depends on definitions: df-bi 116 df-3an 947 df-tru 1317 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ral 2395 df-rex 2396 df-v 2659 df-un 3041 df-in 3043 df-ss 3050 df-pw 3478 df-sn 3499 df-pr 3500 df-op 3502 df-uni 3703 df-br 3896 df-opab 3950 df-xp 4505 df-rel 4506 df-cnv 4507 df-dm 4509 df-rn 4510 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |