Theorem mptrcl 5496

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 5030	. 2 |- dom F C_ A
3	1	funmpt2 5157	. . . 4 |- Fun F
4		funrel 5135	. . . 4 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . 3 |- Rel F
6		relelfvdm 5446	. . 3 |- ( ( Rel F /\ I e. ( F ` X ) ) -> X e. dom F )
7	5, 6	mpan 420	. 2 |- ( I e. ( F ` X ) -> X e. dom F )
8	2, 7	sseldi 3090	1 |- ( I e. ( F ` X ) -> X e. A )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   |-> cmpt 3984   dom cdm 4534   Rel wrel 4539   Fun wfun 5112   ` cfv 5118

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fv 5126

This theorem is referenced by:  psmetdmdm  12482  psmetf  12483  psmet0  12485  psmettri2  12486  psmetres2  12491