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Theorem eldmrexrnb 6849
 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 6351 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 6351 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 6848 . . 3 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
21adantr 484 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
3 eleq1 2903 . . . . 5 (𝑥 = (𝐹𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹𝑌) ∈ ran 𝐹))
4 elnelne2 3129 . . . . . . . . 9 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹𝑌) ≠ ∅)
5 n0 4293 . . . . . . . . . 10 ((𝐹𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑌))
6 elfvdm 6693 . . . . . . . . . . 11 (𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
76exlimiv 1932 . . . . . . . . . 10 (∃𝑦 𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
85, 7sylbi 220 . . . . . . . . 9 ((𝐹𝑌) ≠ ∅ → 𝑌 ∈ dom 𝐹)
94, 8syl 17 . . . . . . . 8 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹)
109expcom 417 . . . . . . 7 (∅ ∉ ran 𝐹 → ((𝐹𝑌) ∈ ran 𝐹𝑌 ∈ dom 𝐹))
1110adantl 485 . . . . . 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 256 . . . 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 3275 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌) → 𝑌 ∈ dom 𝐹))
162, 15impbid 215 1 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115   ≠ wne 3014   ∉ wnel 3118  ∃wrex 3134  ∅c0 4276  dom cdm 5542  ran crn 5543  Fun wfun 6337  ‘cfv 6343 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-fv 6351 This theorem is referenced by: (None)
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