Theorem mptrcl 5641

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

Step	Hyp	Ref	Expression
1		fvmpt2.1	. . 3 |- F = ( x e. A |-> B )
2	1	dmmptss 5163	. 2 |- dom F C_ A
3	1	funmpt2 5294	. . . 4 |- Fun F
4		funrel 5272	. . . 4 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . 3 |- Rel F
6		relelfvdm 5587	. . 3 |- ( ( Rel F /\ I e. ( F ` X ) ) -> X e. dom F )
7	5, 6	mpan 424	. 2 |- ( I e. ( F ` X ) -> X e. dom F )
8	2, 7	sselid 3178	1 |- ( I e. ( F ` X ) -> X e. A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   |-> cmpt 4091   dom cdm 4660   Rel wrel 4665   Fun wfun 5249   ` cfv 5255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fv 5263

This theorem is referenced by:  divsfval  12914  submrcl  13046  issubg  13246  isnsg  13275  issubrng  13698  issubrg  13720  zrhval  14116  psmetdmdm  14503  psmetf  14504  psmet0  14506  psmettri2  14507  psmetres2  14512  plybss  14912