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Theorem eldmrexrnb 7088
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 6547 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 6547 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 7087 . . 3 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
21adantr 482 . 2 ((Fun 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
3 eleq1 2822 . . . . 5 (𝑥 = (𝐹𝑌) → (𝑥 ∈ ran 𝐹 ↔ (𝐹𝑌) ∈ ran 𝐹))
4 elnelne2 3059 . . . . . . . . 9 (((𝐹𝑌) ∈ ran 𝐹 ∧ ∅ ∉ ran 𝐹) → (𝐹𝑌) ≠ ∅)
5 n0 4344 . . . . . . . . . 10 ((𝐹𝑌) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ (𝐹𝑌))
6 elfvdm 6924 . . . . . . . . . . 11 (𝑦 ∈ (𝐹𝑌) → 𝑌 ∈ dom 𝐹)
76exlimiv 1934 . . . . . . . . . 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 415 . . . . . . 7 (∅ ∉ ran 𝐹 → ((𝐹𝑌) ∈ ran 𝐹𝑌 ∈ dom 𝐹))
1110adantl 483 . . . . . 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 253 . . . 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 3154 . 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 397   = wceq 1542  wex 1782  wcel 2107  wne 2941  wnel 3047  wrex 3071  c0 4320  dom cdm 5674  ran crn 5675  Fun wfun 6533  cfv 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5297  ax-nul 5304  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-br 5147  df-opab 5209  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6491  df-fun 6541  df-fn 6542  df-fv 6547
This theorem is referenced by: (None)
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