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Theorem fvmptss2 5247
Description: A mapping always evaluates to a subset of the substituted expression in the mapping, even if this is a proper class, or we are out of the domain. (Contributed by Mario Carneiro, 13-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2019.)
Hypotheses
Ref Expression
fvmptss2.1  |-  ( x  =  D  ->  B  =  C )
fvmptss2.2  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
fvmptss2  |-  ( F `
 D )  C_  C
Distinct variable groups:    x, A    x, C    x, D
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem fvmptss2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fvss 5189 . 2  |-  ( A. y ( D F y  ->  y  C_  C )  ->  ( F `  D )  C_  C )
2 fvmptss2.2 . . . . . 6  |-  F  =  ( x  e.  A  |->  B )
32funmpt2 4939 . . . . 5  |-  Fun  F
4 funrel 4919 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
53, 4ax-mp 7 . . . 4  |-  Rel  F
65brrelexi 4384 . . 3  |-  ( D F y  ->  D  e.  _V )
7 nfcv 2178 . . . 4  |-  F/_ x D
8 nfmpt1 3850 . . . . . . 7  |-  F/_ x
( x  e.  A  |->  B )
92, 8nfcxfr 2175 . . . . . 6  |-  F/_ x F
10 nfcv 2178 . . . . . 6  |-  F/_ x
y
117, 9, 10nfbr 3808 . . . . 5  |-  F/ x  D F y
12 nfv 1421 . . . . 5  |-  F/ x  y  C_  C
1311, 12nfim 1464 . . . 4  |-  F/ x
( D F y  ->  y  C_  C
)
14 breq1 3767 . . . . 5  |-  ( x  =  D  ->  (
x F y  <->  D F
y ) )
15 fvmptss2.1 . . . . . 6  |-  ( x  =  D  ->  B  =  C )
1615sseq2d 2973 . . . . 5  |-  ( x  =  D  ->  (
y  C_  B  <->  y  C_  C ) )
1714, 16imbi12d 223 . . . 4  |-  ( x  =  D  ->  (
( x F y  ->  y  C_  B
)  <->  ( D F y  ->  y  C_  C ) ) )
18 df-br 3765 . . . . 5  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
19 opabid 3994 . . . . . . 7  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  <->  ( x  e.  A  /\  y  =  B ) )
20 eqimss 2997 . . . . . . . 8  |-  ( y  =  B  ->  y  C_  B )
2120adantl 262 . . . . . . 7  |-  ( ( x  e.  A  /\  y  =  B )  ->  y  C_  B )
2219, 21sylbi 114 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  ->  y 
C_  B )
23 df-mpt 3820 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
242, 23eqtri 2060 . . . . . 6  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
2522, 24eleq2s 2132 . . . . 5  |-  ( <.
x ,  y >.  e.  F  ->  y  C_  B )
2618, 25sylbi 114 . . . 4  |-  ( x F y  ->  y  C_  B )
277, 13, 17, 26vtoclgf 2612 . . 3  |-  ( D  e.  _V  ->  ( D F y  ->  y  C_  C ) )
286, 27mpcom 32 . 2  |-  ( D F y  ->  y  C_  C )
291, 28mpg 1340 1  |-  ( F `
 D )  C_  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   _Vcvv 2557    C_ wss 2917   <.cop 3378   class class class wbr 3764   {copab 3817    |-> cmpt 3818   Rel wrel 4350   Fun wfun 4896   ` cfv 4902
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-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
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-v 2559  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-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-iota 4867  df-fun 4904  df-fv 4910
This theorem is referenced by:  mptfvex  5256
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