MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fvn0fvelrnOLD Structured version   Visualization version   GIF version

Theorem fvn0fvelrnOLD 7110
Description: Obsolete version of fvn0fvelrn 6874 as of 13-Jan-2025. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
fvn0fvelrnOLD ((𝐹𝑋) ≠ ∅ → (𝐹𝑋) ∈ ran 𝐹)

Proof of Theorem fvn0fvelrnOLD
StepHypRef Expression
1 fvfundmfvn0 6886 . 2 ((𝐹𝑋) ≠ ∅ → (𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})))
2 eldmressnsn 5981 . . . 4 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))
3 fvelrn 7028 . . . . . . 7 ((Fun (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) → ((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))
4 pm3.2 471 . . . . . . 7 (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)))
53, 4syl 17 . . . . . 6 ((Fun (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom (𝐹 ↾ {𝑋})) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)))
65ex 414 . . . . 5 (Fun (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))))
76com13 88 . . . 4 (𝑋 ∈ dom 𝐹 → (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → (Fun (𝐹 ↾ {𝑋}) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))))
82, 7mpd 15 . . 3 (𝑋 ∈ dom 𝐹 → (Fun (𝐹 ↾ {𝑋}) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹)))
98imp 408 . 2 ((𝑋 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝑋})) → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹))
10 fvressn 7109 . . . . 5 (𝑋 ∈ dom 𝐹 → ((𝐹 ↾ {𝑋})‘𝑋) = (𝐹𝑋))
1110eleq1d 2819 . . . 4 (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ↔ (𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋})))
12 fvrnressn 7108 . . . 4 (𝑋 ∈ dom 𝐹 → ((𝐹𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
1311, 12sylbid 239 . . 3 (𝑋 ∈ dom 𝐹 → (((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹𝑋) ∈ ran 𝐹))
1413impcom 409 . 2 ((((𝐹 ↾ {𝑋})‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) ∧ 𝑋 ∈ dom 𝐹) → (𝐹𝑋) ∈ ran 𝐹)
151, 9, 143syl 18 1 ((𝐹𝑋) ≠ ∅ → (𝐹𝑋) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  wne 2940  c0 4283  {csn 4587  dom cdm 5634  ran crn 5635  cres 5636  Fun wfun 6491  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator