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Mirrors > Home > ILE Home > Th. List > dmmpt2ssx | Unicode version |
Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.) |
Ref | Expression |
---|---|
fmpt2x.1 |
Ref | Expression |
---|---|
dmmpt2ssx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2178 | . . . . 5 | |
2 | nfcsb1v 2882 | . . . . 5 | |
3 | nfcv 2178 | . . . . 5 | |
4 | nfcv 2178 | . . . . 5 | |
5 | nfcsb1v 2882 | . . . . 5 | |
6 | nfcv 2178 | . . . . . 6 | |
7 | nfcsb1v 2882 | . . . . . 6 | |
8 | 6, 7 | nfcsb 2884 | . . . . 5 |
9 | csbeq1a 2860 | . . . . 5 | |
10 | csbeq1a 2860 | . . . . . 6 | |
11 | csbeq1a 2860 | . . . . . 6 | |
12 | 10, 11 | sylan9eqr 2094 | . . . . 5 |
13 | 1, 2, 3, 4, 5, 8, 9, 12 | cbvmpt2x 5582 | . . . 4 |
14 | fmpt2x.1 | . . . 4 | |
15 | vex 2560 | . . . . . . . 8 | |
16 | vex 2560 | . . . . . . . 8 | |
17 | 15, 16 | op1std 5775 | . . . . . . 7 |
18 | 17 | csbeq1d 2858 | . . . . . 6 |
19 | 15, 16 | op2ndd 5776 | . . . . . . . 8 |
20 | 19 | csbeq1d 2858 | . . . . . . 7 |
21 | 20 | csbeq2dv 2875 | . . . . . 6 |
22 | 18, 21 | eqtrd 2072 | . . . . 5 |
23 | 22 | mpt2mptx 5595 | . . . 4 |
24 | 13, 14, 23 | 3eqtr4i 2070 | . . 3 |
25 | 24 | dmmptss 4817 | . 2 |
26 | nfcv 2178 | . . 3 | |
27 | nfcv 2178 | . . . 4 | |
28 | 27, 2 | nfxp 4371 | . . 3 |
29 | sneq 3386 | . . . 4 | |
30 | 29, 9 | xpeq12d 4370 | . . 3 |
31 | 26, 28, 30 | cbviun 3694 | . 2 |
32 | 25, 31 | sseqtr4i 2978 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1243 csb 2852 wss 2917 csn 3375 cop 3378 ciun 3657 cmpt 3818 cxp 4343 cdm 4345 cfv 4902 cmpt2 5514 c1st 5765 c2nd 5766 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fv 4910 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: mpt2exxg 5833 mpt2xopn0yelv 5854 |
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