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Theorem elrnrexdm 6271
 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, 8-Dec-2017.)
Assertion
Ref Expression
elrnrexdm (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem elrnrexdm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2611 . . . . . 6 (𝑌 ∈ ran 𝐹𝑌 = 𝑌)
21ancli 572 . . . . 5 (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
32adantl 481 . . . 4 ((Fun 𝐹𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
4 eqeq2 2621 . . . . 5 (𝑦 = 𝑌 → (𝑌 = 𝑦𝑌 = 𝑌))
54rspcev 3282 . . . 4 ((𝑌 ∈ ran 𝐹𝑌 = 𝑌) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
63, 5syl 17 . . 3 ((Fun 𝐹𝑌 ∈ ran 𝐹) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
76ex 449 . 2 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦))
8 funfn 5833 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
9 eqeq2 2621 . . . 4 (𝑦 = (𝐹𝑥) → (𝑌 = 𝑦𝑌 = (𝐹𝑥)))
109rexrn 6269 . . 3 (𝐹 Fn dom 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
118, 10sylbi 206 . 2 (Fun 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
127, 11sylibd 228 1 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = 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:  nbgraf1olem1  25970  vdusgra0nedg  26435  usgravd0nedg  26445  bj-toprntopon  32244  bj-ccinftydisj  32277  gneispace  37452  1wlkiswwlksupgr2  41074
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