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Mirrors > Home > MPE Home > Th. List > dmsnopg | Structured version Visualization version Unicode version |
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnopg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3203 | . . . . . 6 | |
2 | vex 3203 | . . . . . 6 | |
3 | 1, 2 | opth1 4944 | . . . . 5 |
4 | 3 | exlimiv 1858 | . . . 4 |
5 | opeq1 4402 | . . . . 5 | |
6 | opeq2 4403 | . . . . . . 7 | |
7 | 6 | eqeq1d 2624 | . . . . . 6 |
8 | 7 | spcegv 3294 | . . . . 5 |
9 | 5, 8 | syl5 34 | . . . 4 |
10 | 4, 9 | impbid2 216 | . . 3 |
11 | 1 | eldm2 5322 | . . . 4 |
12 | opex 4932 | . . . . . 6 | |
13 | 12 | elsn 4192 | . . . . 5 |
14 | 13 | exbii 1774 | . . . 4 |
15 | 11, 14 | bitri 264 | . . 3 |
16 | velsn 4193 | . . 3 | |
17 | 10, 15, 16 | 3bitr4g 303 | . 2 |
18 | 17 | eqrdv 2620 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wex 1704 wcel 1990 csn 4177 cop 4183 cdm 5114 |
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-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 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-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-dm 5124 |
This theorem is referenced by: dmsnopss 5607 dmpropg 5608 dmsnop 5609 rnsnopg 5614 fnsng 5938 funprg 5940 funprgOLD 5941 funtpg 5942 funtpgOLD 5943 fntpg 5948 suppsnop 7309 funsnfsupp 8299 s1dmALT 13389 setsval 15888 setsdm 15892 estrreslem2 16778 snstriedgval 25930 1loopgrvd0 26400 1hevtxdg0 26401 1hevtxdg1 26402 1egrvtxdg1 26405 p1evtxdeqlem 26408 wlkp1 26578 eupthp1 27076 trlsegvdeglem5 27084 bnj96 30935 bnj535 30960 ovnovollem1 40870 |
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