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Theorem mptrcl 7025
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.)
Hypothesis
Ref Expression
mptrcl.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 n0i 4346 . 2 (𝐼 ∈ (𝐹𝑋) → ¬ (𝐹𝑋) = ∅)
2 mptrcl.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
32dmmptss 6263 . . . 4 dom 𝐹𝐴
43sseli 3991 . . 3 (𝑋 ∈ dom 𝐹𝑋𝐴)
5 ndmfv 6942 . . 3 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
64, 5nsyl4 158 . 2 (¬ (𝐹𝑋) = ∅ → 𝑋𝐴)
71, 6syl 17 1 (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  c0 4339  cmpt 5231  dom cdm 5689  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571
This theorem is referenced by:  bitsval  16458  subcrcl  17864  initorcl  18044  termorcl  18045  zeroorcl  18046  submrcl  18828  issubg  19157  isnsg  19186  issubrng  20564  issubrg  20588  issdrg  20806  abvrcl  20831  isobs  21758  mhprcl  22165  islocfin  23541  kgeni  23561  elmptrab  23851  isphtpc  25040  cfili  25316  cfilfcls  25322  plybss  26248  eleenn  28926  neircl  48701
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