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Mirrors > Home > MPE Home > Th. List > dfdm2 | Structured version Visualization version GIF version |
Description: Alternate definition of domain df-dm 5352 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 5540 | . . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴) | |
2 | cocnvcnv2 5888 | . . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴) | |
3 | 1, 2 | eqtri 2849 | . . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴) |
4 | 3 | unieqi 4667 | . . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴) |
5 | 4 | unieqi 4667 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴) |
6 | unidmrn 5906 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) | |
7 | 5, 6 | eqtr3i 2851 | . 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) |
8 | df-rn 5353 | . . . . 5 ⊢ ran 𝐴 = dom ◡𝐴 | |
9 | 8 | eqcomi 2834 | . . . 4 ⊢ dom ◡𝐴 = ran 𝐴 |
10 | dmcoeq 5621 | . . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴 |
12 | rncoeq 5622 | . . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴) | |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴 |
14 | dfdm4 5548 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
15 | 13, 14 | eqtr4i 2852 | . . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴 |
16 | 11, 15 | uneq12i 3992 | . 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴) |
17 | unidm 3983 | . 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴 | |
18 | 7, 16, 17 | 3eqtrri 2854 | 1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1658 ∪ cun 3796 ∪ cuni 4658 ◡ccnv 5341 dom cdm 5342 ran crn 5343 ∘ ccom 5346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 |
This theorem is referenced by: (None) |
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