MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mptrcl Structured version   Visualization version   GIF version

Theorem mptrcl 6478
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 4084 . 2 (𝐼 ∈ (𝐹𝑋) → ¬ (𝐹𝑋) = ∅)
2 mptrcl.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
32dmmptss 5817 . . . 4 dom 𝐹𝐴
43sseli 3757 . . 3 (𝑋 ∈ dom 𝐹𝑋𝐴)
5 ndmfv 6405 . . 3 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
64, 5nsyl4 157 . 2 (¬ (𝐹𝑋) = ∅ → 𝑋𝐴)
71, 6syl 17 1 (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1652  wcel 2155  c0 4079  cmpt 4888  dom cdm 5277  cfv 6068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fv 6076
This theorem is referenced by:  initorcl  16911  termorcl  16912  zeroorcl  16913  issubrg  19049  elmptrab  21910
  Copyright terms: Public domain W3C validator