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Theorem mptrcl 5469
Description: Reverse closure for a mapping: If the function value of a mapping has a member, the argument belongs to the base class of the mapping. (Contributed by AV, 4-Apr-2020.) (Revised by Jim Kingdon, 27-Mar-2023.)
Hypothesis
Ref Expression
fvmpt2.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
mptrcl  |-  ( I  e.  ( F `  X )  ->  X  e.  A )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    F( x)    I( x)    X( x)

Proof of Theorem mptrcl
StepHypRef Expression
1 fvmpt2.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
21dmmptss 5003 . 2  |-  dom  F  C_  A
31funmpt2 5130 . . . 4  |-  Fun  F
4 funrel 5108 . . . 4  |-  ( Fun 
F  ->  Rel  F )
53, 4ax-mp 5 . . 3  |-  Rel  F
6 relelfvdm 5419 . . 3  |-  ( ( Rel  F  /\  I  e.  ( F `  X
) )  ->  X  e.  dom  F )
75, 6mpan 418 . 2  |-  ( I  e.  ( F `  X )  ->  X  e.  dom  F )
82, 7sseldi 3063 1  |-  ( I  e.  ( F `  X )  ->  X  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314    e. wcel 1463    |-> cmpt 3957   dom cdm 4507   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-rab 2400  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-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fv 5099
This theorem is referenced by:  psmetdmdm  12388  psmetf  12389  psmet0  12391  psmettri2  12392  psmetres2  12397
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