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Mirrors > Home > MPE Home > Th. List > dfdm2 | Structured version Visualization version GIF version |
Description: Alternate definition of domain df-dm 5124 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 5308 | . . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴) | |
2 | cocnvcnv2 5647 | . . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴) | |
3 | 1, 2 | eqtri 2644 | . . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴) |
4 | 3 | unieqi 4445 | . . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴) |
5 | 4 | unieqi 4445 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴) |
6 | unidmrn 5665 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) | |
7 | 5, 6 | eqtr3i 2646 | . 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) |
8 | df-rn 5125 | . . . . 5 ⊢ ran 𝐴 = dom ◡𝐴 | |
9 | 8 | eqcomi 2631 | . . . 4 ⊢ dom ◡𝐴 = ran 𝐴 |
10 | dmcoeq 5388 | . . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴 |
12 | rncoeq 5389 | . . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴) | |
13 | 9, 12 | ax-mp 5 | . . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴 |
14 | dfdm4 5316 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
15 | 13, 14 | eqtr4i 2647 | . . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴 |
16 | 11, 15 | uneq12i 3765 | . 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴) |
17 | unidm 3756 | . 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴 | |
18 | 7, 16, 17 | 3eqtrri 2649 | 1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∪ cun 3572 ∪ cuni 4436 ◡ccnv 5113 dom cdm 5114 ran crn 5115 ∘ ccom 5118 |
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-co 5123 df-dm 5124 df-rn 5125 df-res 5126 |
This theorem is referenced by: (None) |
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