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Theorem mptrcl 5515
 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 5047 . 2 dom 𝐹𝐴
31funmpt2 5174 . . . 4 Fun 𝐹
4 funrel 5152 . . . 4 (Fun 𝐹 → Rel 𝐹)
53, 4ax-mp 5 . . 3 Rel 𝐹
6 relelfvdm 5465 . . 3 ((Rel 𝐹𝐼 ∈ (𝐹𝑋)) → 𝑋 ∈ dom 𝐹)
75, 6mpan 421 . 2 (𝐼 ∈ (𝐹𝑋) → 𝑋 ∈ dom 𝐹)
82, 7sseldi 3102 1 (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481   ↦ cmpt 3999  dom cdm 4551  Rel wrel 4556  Fun wfun 5129  ‘cfv 5135 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-rab 2427  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-uni 3747  df-br 3940  df-opab 4000  df-mpt 4001  df-id 4226  df-xp 4557  df-rel 4558  df-cnv 4559  df-co 4560  df-dm 4561  df-rn 4562  df-res 4563  df-ima 4564  df-iota 5100  df-fun 5137  df-fv 5143 This theorem is referenced by:  psmetdmdm  12568  psmetf  12569  psmet0  12571  psmettri2  12572  psmetres2  12577
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