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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 5503 | . . . 4 ⊢ Rel ◡𝐴 | |
2 | relfld 5661 | . . . 4 ⊢ (Rel ◡𝐴 → ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴)) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ ∪ ∪ ◡𝐴 = (dom ◡𝐴 ∪ ran ◡𝐴) |
4 | 3 | equncomi 3759 | . 2 ⊢ ∪ ∪ ◡𝐴 = (ran ◡𝐴 ∪ dom ◡𝐴) |
5 | dfdm4 5316 | . . 3 ⊢ dom 𝐴 = ran ◡𝐴 | |
6 | df-rn 5125 | . . 3 ⊢ ran 𝐴 = dom ◡𝐴 | |
7 | 5, 6 | uneq12i 3765 | . 2 ⊢ (dom 𝐴 ∪ ran 𝐴) = (ran ◡𝐴 ∪ dom ◡𝐴) |
8 | 4, 7 | eqtr4i 2647 | 1 ⊢ ∪ ∪ ◡𝐴 = (dom 𝐴 ∪ ran 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∪ cun 3572 ∪ cuni 4436 ◡ccnv 5113 dom cdm 5114 ran crn 5115 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 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: relcnvfld 5666 dfdm2 5667 |
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