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Theorem dmmpossx 6137
 Description: The domain of a mapping is a subset of its base class. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
fmpox.1
Assertion
Ref Expression
dmmpossx
Distinct variable groups:   ,,   ,
Allowed substitution hints:   ()   (,)   (,)

Proof of Theorem dmmpossx
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2296 . . . . 5
2 nfcsb1v 3060 . . . . 5
3 nfcv 2296 . . . . 5
4 nfcv 2296 . . . . 5
5 nfcsb1v 3060 . . . . 5
6 nfcv 2296 . . . . . 6
7 nfcsb1v 3060 . . . . . 6
86, 7nfcsb 3064 . . . . 5
9 csbeq1a 3036 . . . . 5
10 csbeq1a 3036 . . . . . 6
11 csbeq1a 3036 . . . . . 6
1210, 11sylan9eqr 2209 . . . . 5
131, 2, 3, 4, 5, 8, 9, 12cbvmpox 5889 . . . 4
14 fmpox.1 . . . 4
15 vex 2712 . . . . . . . 8
16 vex 2712 . . . . . . . 8
1715, 16op1std 6086 . . . . . . 7
1817csbeq1d 3034 . . . . . 6
1915, 16op2ndd 6087 . . . . . . . 8
2019csbeq1d 3034 . . . . . . 7
2120csbeq2dv 3053 . . . . . 6
2218, 21eqtrd 2187 . . . . 5
2322mpomptx 5902 . . . 4
2413, 14, 233eqtr4i 2185 . . 3
2524dmmptss 5075 . 2
26 nfcv 2296 . . 3
27 nfcv 2296 . . . 4
2827, 2nfxp 4606 . . 3
29 sneq 3567 . . . 4
3029, 9xpeq12d 4604 . . 3
3126, 28, 30cbviun 3882 . 2
3225, 31sseqtrri 3159 1
 Colors of variables: wff set class Syntax hints:   wceq 1332  csb 3027   wss 3098  csn 3556  cop 3559  ciun 3845   cmpt 4021   cxp 4577   cdm 4579  cfv 5163   cmpo 5816  c1st 6076  c2nd 6077 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fv 5171  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079 This theorem is referenced by:  mpoexxg  6148  mpoxopn0yelv  6176
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