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Theorem dmmpt2ssx 5825
 Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
fmpt2x.1
Assertion
Ref Expression
dmmpt2ssx
Distinct variable groups:   ,,   ,
Allowed substitution hints:   ()   (,)   (,)

Proof of Theorem dmmpt2ssx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef 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
86, 7nfcsb 2884 . . . . 5
9 csbeq1a 2860 . . . . 5
10 csbeq1a 2860 . . . . . 6
11 csbeq1a 2860 . . . . . 6
1210, 11sylan9eqr 2094 . . . . 5
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 5582 . . . 4
14 fmpt2x.1 . . . 4
15 vex 2560 . . . . . . . 8
16 vex 2560 . . . . . . . 8
1715, 16op1std 5775 . . . . . . 7
1817csbeq1d 2858 . . . . . 6
1915, 16op2ndd 5776 . . . . . . . 8
2019csbeq1d 2858 . . . . . . 7
2120csbeq2dv 2875 . . . . . 6
2218, 21eqtrd 2072 . . . . 5
2322mpt2mptx 5595 . . . 4
2413, 14, 233eqtr4i 2070 . . 3
2524dmmptss 4817 . 2
26 nfcv 2178 . . 3
27 nfcv 2178 . . . 4
2827, 2nfxp 4371 . . 3
29 sneq 3386 . . . 4
3029, 9xpeq12d 4370 . . 3
3126, 28, 30cbviun 3694 . 2
3225, 31sseqtr4i 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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