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

Theorem mptrcl 6247
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 3901 . 2 (𝐼 ∈ (𝐹𝑋) → ¬ (𝐹𝑋) = ∅)
2 mptrcl.1 . . . . 5 𝐹 = (𝑥𝐴𝐵)
32dmmptss 5593 . . . 4 dom 𝐹𝐴
43sseli 3584 . . 3 (𝑋 ∈ dom 𝐹𝑋𝐴)
5 ndmfv 6176 . . 3 𝑋 ∈ dom 𝐹 → (𝐹𝑋) = ∅)
64, 5nsyl4 156 . 2 (¬ (𝐹𝑋) = ∅ → 𝑋𝐴)
71, 6syl 17 1 (𝐼 ∈ (𝐹𝑋) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1992  c0 3896  cmpt 4678  dom cdm 5079  cfv 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fv 5858
This theorem is referenced by:  initorcl  16560  termorcl  16561  zeroorcl  16562  issubrg  18696  elmptrab  21535
  Copyright terms: Public domain W3C validator