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Mirrors > Home > MPE Home > Th. List > dfdm2 | Structured version Visualization version GIF version |
Description: Alternate definition of domain df-dm 5699 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 5899 | . . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴) | |
2 | cocnvcnv2 6280 | . . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴) | |
3 | 1, 2 | eqtri 2763 | . . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴) |
4 | 3 | unieqi 4924 | . . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴) |
5 | 4 | unieqi 4924 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴) |
6 | unidmrn 6301 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) | |
7 | 5, 6 | eqtr3i 2765 | . 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) |
8 | df-rn 5700 | . . . . 5 ⊢ ran 𝐴 = dom ◡𝐴 | |
9 | 8 | eqcomi 2744 | . . . 4 ⊢ dom ◡𝐴 = ran 𝐴 |
10 | dmcoeq 5992 | . . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴 |
12 | rncoeq 5993 | . . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴) | |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴 |
14 | dfdm4 5909 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
15 | 13, 14 | eqtr4i 2766 | . . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴 |
16 | 11, 15 | uneq12i 4176 | . 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴) |
17 | unidm 4167 | . 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴 | |
18 | 7, 16, 17 | 3eqtrri 2768 | 1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3961 ∪ cuni 4912 ◡ccnv 5688 dom cdm 5689 ran crn 5690 ∘ ccom 5693 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 |
This theorem is referenced by: (None) |
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