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| Mirrors > Home > MPE Home > Th. List > dfdm2 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of domain df-dm 5695 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
| Ref | Expression |
|---|---|
| dfdm2 | ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cnvco 5896 | . . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴) | |
| 2 | cocnvcnv2 6278 | . . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴) | |
| 3 | 1, 2 | eqtri 2765 | . . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴) |
| 4 | 3 | unieqi 4919 | . . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴) |
| 5 | 4 | unieqi 4919 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴) |
| 6 | unidmrn 6299 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) | |
| 7 | 5, 6 | eqtr3i 2767 | . 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) |
| 8 | df-rn 5696 | . . . . 5 ⊢ ran 𝐴 = dom ◡𝐴 | |
| 9 | 8 | eqcomi 2746 | . . . 4 ⊢ dom ◡𝐴 = ran 𝐴 |
| 10 | dmcoeq 5989 | . . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴) | |
| 11 | 9, 10 | ax-mp 5 | . . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴 |
| 12 | rncoeq 5990 | . . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴) | |
| 13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴 |
| 14 | dfdm4 5906 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
| 15 | 13, 14 | eqtr4i 2768 | . . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴 |
| 16 | 11, 15 | uneq12i 4166 | . 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴) |
| 17 | unidm 4157 | . 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴 | |
| 18 | 7, 16, 17 | 3eqtrri 2770 | 1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∪ cun 3949 ∪ cuni 4907 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ∘ ccom 5689 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 |
| This theorem is referenced by: (None) |
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