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Theorem mptrcl 5619
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 = (𝑥𝐴𝐵)
Assertion
Ref Expression
mptrcl (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
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 = (𝑥𝐴𝐵)
21dmmptss 5143 . 2 dom 𝐹𝐴
31funmpt2 5274 . . . 4 Fun 𝐹
4 funrel 5252 . . . 4 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 5 . . 3 Rel 𝐹
6 relelfvdm 5566 . . 3 ((Rel 𝐹𝐼 ∈ (𝐹𝑋)) → 𝑋 ∈ dom 𝐹)
75, 6mpan 424 . 2 (𝐼 ∈ (𝐹𝑋) → 𝑋 ∈ dom 𝐹)
82, 7sselid 3168 1 (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  cmpt 4079  dom cdm 4644  Rel wrel 4649  Fun wfun 5229  cfv 5235
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 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fv 5243
This theorem is referenced by:  submrcl  12938  issubg  13129  isnsg  13158  issubrng  13563  issubrg  13585  psmetdmdm  14301  psmetf  14302  psmet0  14304  psmettri2  14305  psmetres2  14310
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