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Theorem fvmptss2 5275
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 5217 . 2  |-  ( A. y ( D F y  ->  y  C_  C )  ->  ( F `  D )  C_  C )
2 fvmptss2.2 . . . . . 6  |-  F  =  ( x  e.  A  |->  B )
32funmpt2 4967 . . . . 5  |-  Fun  F
4 funrel 4947 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
53, 4ax-mp 7 . . . 4  |-  Rel  F
65brrelexi 4412 . . 3  |-  ( D F y  ->  D  e.  _V )
7 nfcv 2194 . . . 4  |-  F/_ x D
8 nfmpt1 3878 . . . . . . 7  |-  F/_ x
( x  e.  A  |->  B )
92, 8nfcxfr 2191 . . . . . 6  |-  F/_ x F
10 nfcv 2194 . . . . . 6  |-  F/_ x
y
117, 9, 10nfbr 3836 . . . . 5  |-  F/ x  D F y
12 nfv 1437 . . . . 5  |-  F/ x  y  C_  C
1311, 12nfim 1480 . . . 4  |-  F/ x
( D F y  ->  y  C_  C
)
14 breq1 3795 . . . . 5  |-  ( x  =  D  ->  (
x F y  <->  D F
y ) )
15 fvmptss2.1 . . . . . 6  |-  ( x  =  D  ->  B  =  C )
1615sseq2d 3001 . . . . 5  |-  ( x  =  D  ->  (
y  C_  B  <->  y  C_  C ) )
1714, 16imbi12d 227 . . . 4  |-  ( x  =  D  ->  (
( x F y  ->  y  C_  B
)  <->  ( D F y  ->  y  C_  C ) ) )
18 df-br 3793 . . . . 5  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
19 opabid 4022 . . . . . . 7  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  <->  ( x  e.  A  /\  y  =  B ) )
20 eqimss 3025 . . . . . . . 8  |-  ( y  =  B  ->  y  C_  B )
2120adantl 266 . . . . . . 7  |-  ( ( x  e.  A  /\  y  =  B )  ->  y  C_  B )
2219, 21sylbi 118 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  ->  y 
C_  B )
23 df-mpt 3848 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
242, 23eqtri 2076 . . . . . 6  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
2522, 24eleq2s 2148 . . . . 5  |-  ( <.
x ,  y >.  e.  F  ->  y  C_  B )
2618, 25sylbi 118 . . . 4  |-  ( x F y  ->  y  C_  B )
277, 13, 17, 26vtoclgf 2629 . . 3  |-  ( D  e.  _V  ->  ( D F y  ->  y  C_  C ) )
286, 27mpcom 36 . 2  |-  ( D F y  ->  y  C_  C )
291, 28mpg 1356 1  |-  ( F `
 D )  C_  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259    e. wcel 1409   _Vcvv 2574    C_ wss 2945   <.cop 3406   class class class wbr 3792   {copab 3845    |-> cmpt 3846   Rel wrel 4378   Fun wfun 4924   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-iota 4895  df-fun 4932  df-fv 4938
This theorem is referenced by:  mptfvex  5284
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