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Theorem eldmrexrnb 6683
 Description: For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 6196 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 6196 of the value of a function, (𝐹‘𝑌) = ∅ may mean that the value of 𝐹 at 𝑌 is the empty set or that 𝐹 is not defined at 𝑌. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
Assertion
Ref Expression
eldmrexrnb ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem eldmrexrnb
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eldmrexrn 6682 . . 3 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
21adantr 473 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
3 eleq1 2853 . . . . 5 (𝑥 = (𝐹𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹𝑌) ∈ ran 𝐹))
4 elnelne2 3084 . . . . . . . . 9 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹𝑌) ≠ ∅)
5 n0 4196 . . . . . . . . . 10 ((𝐹𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑌))
6 elfvdm 6531 . . . . . . . . . . 11 (𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
76exlimiv 1889 . . . . . . . . . 10 (∃𝑦 𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
85, 7sylbi 209 . . . . . . . . 9 ((𝐹𝑌) ≠ ∅ → 𝑌 ∈ dom 𝐹)
94, 8syl 17 . . . . . . . 8 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹)
109expcom 406 . . . . . . 7 (∅ ∉ ran 𝐹 → ((𝐹𝑌) ∈ ran 𝐹𝑌 ∈ dom 𝐹))
1110adantl 474 . . . . . 6 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → ((𝐹𝑌) ∈ ran 𝐹𝑌 ∈ dom 𝐹))
1211com12 32 . . . . 5 ((𝐹𝑌) ∈ ran 𝐹 → ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹))
133, 12syl6bi 245 . . . 4 (𝑥 = (𝐹𝑌) → (𝑥 ∈ ran 𝐹 → ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹)))
1413com13 88 . . 3 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑥 ∈ ran 𝐹 → (𝑥 = (𝐹𝑌) → 𝑌 ∈ dom 𝐹)))
1514rexlimdv 3228 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌) → 𝑌 ∈ dom 𝐹))
162, 15impbid 204 1 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507  ∃wex 1742   ∈ wcel 2050   ≠ wne 2967   ∉ wnel 3073  ∃wrex 3089  ∅c0 4178  dom cdm 5407  ran crn 5408  Fun wfun 6182  ‘cfv 6188 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-nel 3074  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-fv 6196 This theorem is referenced by: (None)
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