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Theorem mptrcl 5600
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 5127 . 2 dom 𝐹𝐴
31funmpt2 5257 . . . 4 Fun 𝐹
4 funrel 5235 . . . 4 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 5 . . 3 Rel 𝐹
6 relelfvdm 5549 . . 3 ((Rel 𝐹𝐼 ∈ (𝐹𝑋)) → 𝑋 ∈ dom 𝐹)
75, 6mpan 424 . 2 (𝐼 ∈ (𝐹𝑋) → 𝑋 ∈ dom 𝐹)
82, 7sselid 3155 1 (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  cmpt 4066  dom cdm 4628  Rel wrel 4633  Fun wfun 5212  cfv 5218
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fv 5226
This theorem is referenced by:  submrcl  12867  issubg  13038  isnsg  13067  issubrg  13347  psmetdmdm  13863  psmetf  13864  psmet0  13866  psmettri2  13867  psmetres2  13872
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