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Mirrors > Home > MPE Home > Th. List > dfdm2 | Unicode version |
Description: Alternate definition of domain df-dm 5014 that doesn't require dummy variables. (Contributed by NM, 2-Aug-2010.) |
Ref | Expression |
---|---|
dfdm2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvco 5193 | . . . . . 6 | |
2 | cocnvcnv2 5524 | . . . . . 6 | |
3 | 1, 2 | eqtri 2486 | . . . . 5 |
4 | 3 | unieqi 4258 | . . . 4 |
5 | 4 | unieqi 4258 | . . 3 |
6 | unidmrn 5542 | . . 3 | |
7 | 5, 6 | eqtr3i 2488 | . 2 |
8 | df-rn 5015 | . . . . 5 | |
9 | 8 | eqcomi 2470 | . . . 4 |
10 | dmcoeq 5270 | . . . 4 | |
11 | 9, 10 | ax-mp 5 | . . 3 |
12 | rncoeq 5271 | . . . . 5 | |
13 | 9, 12 | ax-mp 5 | . . . 4 |
14 | dfdm4 5200 | . . . 4 | |
15 | 13, 14 | eqtr4i 2489 | . . 3 |
16 | 11, 15 | uneq12i 3655 | . 2 |
17 | unidm 3646 | . 2 | |
18 | 7, 16, 17 | 3eqtrri 2491 | 1 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1395 u. cun 3473
U. cuni 4249 `' ccnv 5003 dom cdm 5004
ran crn 5005 o. ccom 5008 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1618 ax-4 1631 ax-5 1704 ax-6 1747 ax-7 1790 ax-9 1822 ax-10 1837 ax-11 1842 ax-12 1854 ax-13 1999 ax-ext 2435 ax-sep 4573 ax-nul 4581 ax-pr 4691 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 975 df-tru 1398 df-ex 1613 df-nf 1617 df-sb 1740 df-eu 2286 df-mo 2287 df-clab 2443 df-cleq 2449 df-clel 2452 df-nfc 2607 df-ne 2654 df-ral 2812 df-rex 2813 df-rab 2816 df-v 3111 df-dif 3478 df-un 3480 df-in 3482 df-ss 3489 df-nul 3785 df-if 3942 df-pw 4014 df-sn 4030 df-pr 4032 df-op 4036 df-uni 4250 df-br 4453 df-opab 4511 df-xp 5010 df-rel 5011 df-cnv 5012 df-co 5013 df-dm 5014 df-rn 5015 df-res 5016 |
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