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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 5960 | . . . 4 ⊢ Rel ◡𝐴 | |
2 | relfld 6119 | . . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴) |
4 | 3 | equncomi 4128 | . 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴) |
5 | dfdm4 5757 | . . 3 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | df-rn 5559 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 5, 6 | uneq12i 4134 | . 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴) |
8 | 4, 7 | eqtr4i 2844 | 1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∪ cun 3931 ∪ cuni 4830 ◡ccnv 5547 dom cdm 5548 ran crn 5549 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-rn 5559 |
This theorem is referenced by: relcnvfld 6124 dfdm2 6125 |
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