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Mirrors > Home > ILE Home > Th. List > dfdm2 | GIF version |
Description: Alternate definition of domain df-dm 4373 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 4538 | . . . . . 6 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ ◡◡𝐴) | |
2 | cocnvcnv2 4852 | . . . . . 6 ⊢ (◡𝐴 ∘ ◡◡𝐴) = (◡𝐴 ∘ 𝐴) | |
3 | 1, 2 | eqtri 2101 | . . . . 5 ⊢ ◡(◡𝐴 ∘ 𝐴) = (◡𝐴 ∘ 𝐴) |
4 | 3 | unieqi 3611 | . . . 4 ⊢ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ (◡𝐴 ∘ 𝐴) |
5 | 4 | unieqi 3611 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = ∪ ∪ (◡𝐴 ∘ 𝐴) |
6 | unidmrn 4870 | . . 3 ⊢ ∪ ∪ ◡(◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) | |
7 | 5, 6 | eqtr3i 2103 | . 2 ⊢ ∪ ∪ (◡𝐴 ∘ 𝐴) = (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) |
8 | df-rn 4374 | . . . . 5 ⊢ ran 𝐴 = dom ◡𝐴 | |
9 | 8 | eqcomi 2085 | . . . 4 ⊢ dom ◡𝐴 = ran 𝐴 |
10 | dmcoeq 4622 | . . . 4 ⊢ (dom ◡𝐴 = ran 𝐴 → dom (◡𝐴 ∘ 𝐴) = dom 𝐴) | |
11 | 9, 10 | ax-mp 7 | . . 3 ⊢ dom (◡𝐴 ∘ 𝐴) = dom 𝐴 |
12 | rncoeq 4623 | . . . . 5 ⊢ (dom ◡𝐴 = ran 𝐴 → ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴) | |
13 | 9, 12 | ax-mp 7 | . . . 4 ⊢ ran (◡𝐴 ∘ 𝐴) = ran ◡𝐴 |
14 | dfdm4 4545 | . . . 4 ⊢ dom 𝐴 = ran ◡𝐴 | |
15 | 13, 14 | eqtr4i 2104 | . . 3 ⊢ ran (◡𝐴 ∘ 𝐴) = dom 𝐴 |
16 | 11, 15 | uneq12i 3124 | . 2 ⊢ (dom (◡𝐴 ∘ 𝐴) ∪ ran (◡𝐴 ∘ 𝐴)) = (dom 𝐴 ∪ dom 𝐴) |
17 | unidm 3115 | . 2 ⊢ (dom 𝐴 ∪ dom 𝐴) = dom 𝐴 | |
18 | 7, 16, 17 | 3eqtrri 2106 | 1 ⊢ dom 𝐴 = ∪ ∪ (◡𝐴 ∘ 𝐴) |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∪ cun 2971 ∪ cuni 3601 ◡ccnv 4362 dom cdm 4363 ran crn 4364 ∘ ccom 4367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 |
This theorem is referenced by: (None) |
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