MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eliman0 Structured version   Visualization version   GIF version

Theorem eliman0 6946
Description: A nonempty function value is an element of the image of the function. (Contributed by Thierry Arnoux, 25-Jun-2019.)
Assertion
Ref Expression
eliman0 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))

Proof of Theorem eliman0
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fvbr0 6935 . . . . 5 (𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅)
2 orcom 871 . . . . 5 ((𝐴𝐹(𝐹𝐴) ∨ (𝐹𝐴) = ∅) ↔ ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴)))
31, 2mpbi 230 . . . 4 ((𝐹𝐴) = ∅ ∨ 𝐴𝐹(𝐹𝐴))
43ori 862 . . 3 (¬ (𝐹𝐴) = ∅ → 𝐴𝐹(𝐹𝐴))
5 breq1 5146 . . . 4 (𝑥 = 𝐴 → (𝑥𝐹(𝐹𝐴) ↔ 𝐴𝐹(𝐹𝐴)))
65rspcev 3622 . . 3 ((𝐴𝐵𝐴𝐹(𝐹𝐴)) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
74, 6sylan2 593 . 2 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
8 fvex 6919 . . 3 (𝐹𝐴) ∈ V
98elima 6083 . 2 ((𝐹𝐴) ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 𝑥𝐹(𝐹𝐴))
107, 9sylibr 234 1 ((𝐴𝐵 ∧ ¬ (𝐹𝐴) = ∅) → (𝐹𝐴) ∈ (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wrex 3070  c0 4333   class class class wbr 5143  cima 5688  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569
This theorem is referenced by:  ovima0  7612  setrec2fun  49211
  Copyright terms: Public domain W3C validator