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

Theorem elrnrexdmb 6272
 Description: For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
Assertion
Ref Expression
elrnrexdmb (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem elrnrexdmb
StepHypRef Expression
1 funfn 5833 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fvelrnb 6153 . . 3 (𝐹 Fn dom 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑌))
31, 2sylbi 206 . 2 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑌))
4 eqcom 2617 . . 3 (𝑌 = (𝐹𝑥) ↔ (𝐹𝑥) = 𝑌)
54rexbii 3023 . 2 (∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥) ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑌)
63, 5syl6bbr 277 1 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  edgiedgb  25798  nbgraop1  25954  uhgrspansubgrlem  40514
 Copyright terms: Public domain W3C validator