Theorem mptrcl 5716

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 5224	. 2 |- dom F C_ A
3	1	funmpt2 5356	. . . 4 |- Fun F
4		funrel 5334	. . . 4 |- ( Fun F -> Rel F )
5	3, 4	ax-mp 5	. . 3 |- Rel F
6		relelfvdm 5658	. . 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 3222	1 |- ( I e. ( F ` X ) -> X e. A )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   |-> cmpt 4144   dom cdm 4718   Rel wrel 4723   Fun wfun 5311   ` cfv 5317

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fv 5325

This theorem is referenced by:  bitsval  12449  divsfval  13356  submrcl  13499  issubg  13705  isnsg  13734  issubrng  14157  issubrg  14179  zrhval  14575  psmetdmdm  14992  psmetf  14993  psmet0  14995  psmettri2  14996  psmetres2  15001  plybss  15401