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Theorem fvmptss2 5462
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 5401 . 2  |-  ( A. y ( D F y  ->  y  C_  C )  ->  ( F `  D )  C_  C )
2 fvmptss2.2 . . . . . 6  |-  F  =  ( x  e.  A  |->  B )
32funmpt2 5130 . . . . 5  |-  Fun  F
4 funrel 5108 . . . . 5  |-  ( Fun 
F  ->  Rel  F )
53, 4ax-mp 5 . . . 4  |-  Rel  F
65brrelex1i 4550 . . 3  |-  ( D F y  ->  D  e.  _V )
7 nfcv 2256 . . . 4  |-  F/_ x D
8 nfmpt1 3989 . . . . . . 7  |-  F/_ x
( x  e.  A  |->  B )
92, 8nfcxfr 2253 . . . . . 6  |-  F/_ x F
10 nfcv 2256 . . . . . 6  |-  F/_ x
y
117, 9, 10nfbr 3942 . . . . 5  |-  F/ x  D F y
12 nfv 1491 . . . . 5  |-  F/ x  y  C_  C
1311, 12nfim 1534 . . . 4  |-  F/ x
( D F y  ->  y  C_  C
)
14 breq1 3900 . . . . 5  |-  ( x  =  D  ->  (
x F y  <->  D F
y ) )
15 fvmptss2.1 . . . . . 6  |-  ( x  =  D  ->  B  =  C )
1615sseq2d 3095 . . . . 5  |-  ( x  =  D  ->  (
y  C_  B  <->  y  C_  C ) )
1714, 16imbi12d 233 . . . 4  |-  ( x  =  D  ->  (
( x F y  ->  y  C_  B
)  <->  ( D F y  ->  y  C_  C ) ) )
18 df-br 3898 . . . . 5  |-  ( x F y  <->  <. x ,  y >.  e.  F
)
19 opabid 4147 . . . . . . 7  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  <->  ( x  e.  A  /\  y  =  B ) )
20 eqimss 3119 . . . . . . . 8  |-  ( y  =  B  ->  y  C_  B )
2120adantl 273 . . . . . . 7  |-  ( ( x  e.  A  /\  y  =  B )  ->  y  C_  B )
2219, 21sylbi 120 . . . . . 6  |-  ( <.
x ,  y >.  e.  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }  ->  y 
C_  B )
23 df-mpt 3959 . . . . . . 7  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
242, 23eqtri 2136 . . . . . 6  |-  F  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
2522, 24eleq2s 2210 . . . . 5  |-  ( <.
x ,  y >.  e.  F  ->  y  C_  B )
2618, 25sylbi 120 . . . 4  |-  ( x F y  ->  y  C_  B )
277, 13, 17, 26vtoclgf 2716 . . 3  |-  ( D  e.  _V  ->  ( D F y  ->  y  C_  C ) )
286, 27mpcom 36 . 2  |-  ( D F y  ->  y  C_  C )
291, 28mpg 1410 1  |-  ( F `
 D )  C_  C
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   _Vcvv 2658    C_ wss 3039   <.cop 3498   class class class wbr 3897   {copab 3956    |-> cmpt 3957   Rel wrel 4512   Fun wfun 5085   ` cfv 5091
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-iota 5056  df-fun 5093  df-fv 5099
This theorem is referenced by:  mptfvex  5472
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