Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > unidmrn | GIF version |
Description: The double union of the converse of a class is its field. (Contributed by NM, 4-Jun-2008.) |
Ref | Expression |
---|---|
unidmrn | ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 4976 | . . . 4 ⊢ Rel ◡𝐴 | |
2 | relfld 5126 | . . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴) |
4 | 3 | equncomi 3263 | . 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴) |
5 | dfdm4 4790 | . . 3 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | df-rn 4609 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 5, 6 | uneq12i 3269 | . 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴) |
8 | 4, 7 | eqtr4i 2188 | 1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∪ cun 3109 ∪ cuni 3783 ◡ccnv 4597 dom cdm 4598 ran crn 4599 Rel wrel 4603 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-xp 4604 df-rel 4605 df-cnv 4606 df-dm 4608 df-rn 4609 |
This theorem is referenced by: relcnvfld 5131 dfdm2 5132 |
Copyright terms: Public domain | W3C validator |