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Theorem eldmrexrnb 6327
 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 5860 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 5860 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 6326 . . 3 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
21adantr 481 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
3 eleq1 2686 . . . . 5 (𝑥 = (𝐹𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹𝑌) ∈ ran 𝐹))
4 elnelne2 2904 . . . . . . . . 9 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹𝑌) ≠ ∅)
5 n0 3912 . . . . . . . . . 10 ((𝐹𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑌))
6 elfvdm 6182 . . . . . . . . . . 11 (𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
76exlimiv 1855 . . . . . . . . . 10 (∃𝑦 𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
85, 7sylbi 207 . . . . . . . . 9 ((𝐹𝑌) ≠ ∅ → 𝑌 ∈ dom 𝐹)
94, 8syl 17 . . . . . . . 8 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹)
109expcom 451 . . . . . . 7 (∅ ∉ ran 𝐹 → ((𝐹𝑌) ∈ ran 𝐹𝑌 ∈ dom 𝐹))
1110adantl 482 . . . . . 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 243 . . . 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 3024 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌) → 𝑌 ∈ dom 𝐹))
162, 15impbid 202 1 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790   ∉ wnel 2893  ∃wrex 2908  ∅c0 3896  dom cdm 5079  ran crn 5080  Fun wfun 5846  ‘cfv 5852 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860 This theorem is referenced by: (None)
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