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Theorem dmmpt2ssx 6417
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  =  ( x  e.  A ,  y  e.  B  |->  C )
Assertion
Ref Expression
dmmpt2ssx  |-  dom  F  C_ 
U_ x  e.  A  ( { x }  X.  B )
Distinct variable groups:    x, y, A    y, B
Allowed substitution hints:    B( x)    C( x, y)    F( x, y)

Proof of Theorem dmmpt2ssx
Dummy variables  u  t  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfcv 2573 . . . . 5  |-  F/_ u B
2 nfcsb1v 3284 . . . . 5  |-  F/_ x [_ u  /  x ]_ B
3 nfcv 2573 . . . . 5  |-  F/_ u C
4 nfcv 2573 . . . . 5  |-  F/_ v C
5 nfcsb1v 3284 . . . . 5  |-  F/_ x [_ u  /  x ]_ [_ v  /  y ]_ C
6 nfcv 2573 . . . . . 6  |-  F/_ y
u
7 nfcsb1v 3284 . . . . . 6  |-  F/_ y [_ v  /  y ]_ C
86, 7nfcsb 3286 . . . . 5  |-  F/_ y [_ u  /  x ]_ [_ v  /  y ]_ C
9 csbeq1a 3260 . . . . 5  |-  ( x  =  u  ->  B  =  [_ u  /  x ]_ B )
10 csbeq1a 3260 . . . . . 6  |-  ( y  =  v  ->  C  =  [_ v  /  y ]_ C )
11 csbeq1a 3260 . . . . . 6  |-  ( x  =  u  ->  [_ v  /  y ]_ C  =  [_ u  /  x ]_ [_ v  /  y ]_ C )
1210, 11sylan9eqr 2491 . . . . 5  |-  ( ( x  =  u  /\  y  =  v )  ->  C  =  [_ u  /  x ]_ [_ v  /  y ]_ C
)
131, 2, 3, 4, 5, 8, 9, 12cbvmpt2x 6151 . . . 4  |-  ( x  e.  A ,  y  e.  B  |->  C )  =  ( u  e.  A ,  v  e. 
[_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
14 fmpt2x.1 . . . 4  |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )
15 vex 2960 . . . . . . . 8  |-  u  e. 
_V
16 vex 2960 . . . . . . . 8  |-  v  e. 
_V
1715, 16op1std 6358 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  ( 1st `  t
)  =  u )
1817csbeq1d 3258 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
1915, 16op2ndd 6359 . . . . . . . 8  |-  ( t  =  <. u ,  v
>.  ->  ( 2nd `  t
)  =  v )
2019csbeq1d 3258 . . . . . . 7  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 2nd `  t
)  /  y ]_ C  =  [_ v  / 
y ]_ C )
2120csbeq2dv 3277 . . . . . 6  |-  ( t  =  <. u ,  v
>.  ->  [_ u  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2218, 21eqtrd 2469 . . . . 5  |-  ( t  =  <. u ,  v
>.  ->  [_ ( 1st `  t
)  /  x ]_ [_ ( 2nd `  t
)  /  y ]_ C  =  [_ u  /  x ]_ [_ v  / 
y ]_ C )
2322mpt2mptx 6165 . . . 4  |-  ( t  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  t )  /  x ]_ [_ ( 2nd `  t )  / 
y ]_ C )  =  ( u  e.  A ,  v  e.  [_ u  /  x ]_ B  |->  [_ u  /  x ]_ [_ v  /  y ]_ C
)
2413, 14, 233eqtr4i 2467 . . 3  |-  F  =  ( t  e.  U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )  |->  [_ ( 1st `  t )  /  x ]_ [_ ( 2nd `  t )  /  y ]_ C )
2524dmmptss 5367 . 2  |-  dom  F  C_ 
U_ u  e.  A  ( { u }  X.  [_ u  /  x ]_ B )
26 nfcv 2573 . . 3  |-  F/_ u
( { x }  X.  B )
27 nfcv 2573 . . . 4  |-  F/_ x { u }
2827, 2nfxp 4905 . . 3  |-  F/_ x
( { u }  X.  [_ u  /  x ]_ B )
29 sneq 3826 . . . 4  |-  ( x  =  u  ->  { x }  =  { u } )
3029, 9xpeq12d 4904 . . 3  |-  ( x  =  u  ->  ( { x }  X.  B )  =  ( { u }  X.  [_ u  /  x ]_ B ) )
3126, 28, 30cbviun 4129 . 2  |-  U_ x  e.  A  ( {
x }  X.  B
)  =  U_ u  e.  A  ( {
u }  X.  [_ u  /  x ]_ B
)
3225, 31sseqtr4i 3382 1  |-  dom  F  C_ 
U_ x  e.  A  ( { x }  X.  B )
Colors of variables: wff set class
Syntax hints:    = wceq 1653   [_csb 3252    C_ wss 3321   {csn 3815   <.cop 3818   U_ciun 4094    e. cmpt 4267    X. cxp 4877   dom cdm 4879   ` cfv 5455    e. cmpt2 6084   1stc1st 6348   2ndc2nd 6349
This theorem is referenced by:  mpt2exxg  6423  mpt2xopn0yelv  6465  mpt2xopxnop0  6467  dmcoass  14222  dvbsss  19790  perfdvf  19791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fv 5463  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351
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