Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > unidmrn | Structured version Visualization version 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 5422 | . . . 4 ⊢ Rel ◡𝐴 | |
2 | relfld 5578 | . . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴) |
4 | 3 | equncomi 3721 | . 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴) |
5 | dfdm4 5238 | . . 3 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | df-rn 5049 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 5, 6 | uneq12i 3727 | . 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴) |
8 | 4, 7 | eqtr4i 2635 | 1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 ∪ cuni 4372 ◡ccnv 5037 dom cdm 5038 ran crn 5039 Rel wrel 5043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: relcnvfld 5583 dfdm2 5584 |
Copyright terms: Public domain | W3C validator |