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Theorem eldmrexrnb 7093
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 6551 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 6551 of the value of a function, (𝐹𝑌) = ∅ 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 7092 . . 3 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
21adantr 480 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
3 eleq1 2820 . . . . 5 (𝑥 = (𝐹𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹𝑌) ∈ ran 𝐹))
4 elnelne2 3057 . . . . . . . . 9 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹𝑌) ≠ ∅)
5 n0 4346 . . . . . . . . . 10 ((𝐹𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑌))
6 elfvdm 6928 . . . . . . . . . . 11 (𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
76exlimiv 1932 . . . . . . . . . 10 (∃𝑦 𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
85, 7sylbi 216 . . . . . . . . 9 ((𝐹𝑌) ≠ ∅ → 𝑌 ∈ dom 𝐹)
94, 8syl 17 . . . . . . . 8 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹)
109expcom 413 . . . . . . 7 (∅ ∉ ran 𝐹 → ((𝐹𝑌) ∈ ran 𝐹𝑌 ∈ dom 𝐹))
1110adantl 481 . . . . . 6 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → ((𝐹𝑌) ∈ ran 𝐹𝑌 ∈ dom 𝐹))
1211com12 32 . . . . 5 ((𝐹𝑌) ∈ ran 𝐹 → ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → 𝑌 ∈ dom 𝐹))
133, 12biimtrdi 252 . . . 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 3152 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌) → 𝑌 ∈ dom 𝐹))
162, 15impbid 211 1 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 ↔ ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1540  wex 1780  wcel 2105  wne 2939  wnel 3045  wrex 3069  c0 4322  dom cdm 5676  ran crn 5677  Fun wfun 6537  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by: (None)
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