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Theorem mptrcl 6769
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 4296 . 2 (𝐼 ∈ (𝐹𝑋) → ¬ (𝐹𝑋) = ∅)
2 mptrcl.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
32dmmptss 6088 . . . 4 dom 𝐹𝐴
43sseli 3960 . . 3 (𝑋 ∈ dom 𝐹𝑋𝐴)
5 ndmfv 6693 . . 3 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
64, 5nsyl4 161 . 2 (¬ (𝐹𝑋) = ∅ → 𝑋𝐴)
71, 6syl 17 1 (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  c0 4288  cmpt 5137  dom cdm 5548  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356
This theorem is referenced by:  bitsval  15761  subcrcl  17074  initorcl  17242  termorcl  17243  zeroorcl  17244  submrcl  17955  issubg  18217  isnsg  18245  issubrg  19464  issdrg  19503  abvrcl  19521  isobs  20792  islocfin  22053  kgeni  22073  elmptrab  22363  isphtpc  23525  cfili  23798  cfilfcls  23804  plybss  24711  eleenn  26609
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