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Theorem dmmpt2ssx 5767
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  F  ,  |->  C
Assertion
Ref Expression
dmmpt2ssx  dom  F  C_ 
U_  { }  X.
Distinct variable groups:   ,,   ,
Allowed substitution hints:   ()    C(,)    F(,)

Proof of Theorem dmmpt2ssx
Dummy variables  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2175 . . . . 5  F/_
2 nfcsb1v 2876 . . . . 5  F/_ [_  ]_
3 nfcv 2175 . . . . 5  F/_ C
4 nfcv 2175 . . . . 5  F/_ C
5 nfcsb1v 2876 . . . . 5  F/_ [_  ]_ [_  ]_ C
6 nfcv 2175 . . . . . 6  F/_
7 nfcsb1v 2876 . . . . . 6  F/_ [_  ]_ C
86, 7nfcsb 2878 . . . . 5  F/_ [_  ]_
[_  ]_ C
9 csbeq1a 2854 . . . . 5  [_  ]_
10 csbeq1a 2854 . . . . . 6  C  [_  ]_ C
11 csbeq1a 2854 . . . . . 6  [_  ]_ C 
[_  ]_
[_  ]_ C
1210, 11sylan9eqr 2091 . . . . 5  C  [_  ]_ [_  ]_ C
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 5524 . . . 4  ,  |->  C  , 
[_  ]_  |->  [_  ]_
[_  ]_ C
14 fmpt2x.1 . . . 4  F  ,  |->  C
15 vex 2554 . . . . . . . 8 
_V
16 vex 2554 . . . . . . . 8 
_V
1715, 16op1std 5717 . . . . . . 7  t  <. , 
>.  1st `  t
1817csbeq1d 2852 . . . . . 6  t  <. , 
>.  [_ 1st `  t  ]_
[_ 2nd `  t  ]_ C  [_  ]_
[_ 2nd `  t  ]_ C
1915, 16op2ndd 5718 . . . . . . . 8  t  <. , 
>.  2nd `  t
2019csbeq1d 2852 . . . . . . 7  t  <. , 
>.  [_ 2nd `  t  ]_ C  [_  ]_ C
2120csbeq2dv 2869 . . . . . 6  t  <. , 
>.  [_  ]_ [_ 2nd `  t  ]_ C  [_  ]_ [_  ]_ C
2218, 21eqtrd 2069 . . . . 5  t  <. , 
>.  [_ 1st `  t  ]_
[_ 2nd `  t  ]_ C  [_  ]_
[_  ]_ C
2322mpt2mptx 5537 . . . 4  t  U_  { }  X.  [_  ]_  |->  [_ 1st `  t  ]_ [_ 2nd `  t  ]_ C  ,  [_  ]_  |->  [_  ]_ [_  ]_ C
2413, 14, 233eqtr4i 2067 . . 3  F  t  U_  { }  X.  [_  ]_  |->  [_ 1st `  t  ]_ [_ 2nd `  t  ]_ C
2524dmmptss 4760 . 2  dom  F  C_ 
U_  { }  X.  [_  ]_
26 nfcv 2175 . . 3  F/_ { }  X.
27 nfcv 2175 . . . 4  F/_ { }
2827, 2nfxp 4314 . . 3  F/_ { }  X.  [_  ]_
29 sneq 3378 . . . 4  { }  { }
3029, 9xpeq12d 4313 . . 3  { }  X.  { }  X.  [_  ]_
3126, 28, 30cbviun 3685 . 2  U_  { }  X. 
U_  { }  X.  [_  ]_
3225, 31sseqtr4i 2972 1  dom  F  C_ 
U_  { }  X.
Colors of variables: wff set class
Syntax hints:   wceq 1242   [_csb 2846    C_ wss 2911   {csn 3367   <.cop 3370   U_ciun 3648    |-> cmpt 3809    X. cxp 4286   dom cdm 4288   ` cfv 4845    |-> cmpt2 5457   1stc1st 5707   2ndc2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fv 4853  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710
This theorem is referenced by:  mpt2exxg  5775  mpt2xopn0yelv  5795
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