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Theorem eldmrexrn 6910
Description: For any element in the domain of a function there is an element in the range of the function which is the function value for the element of the domain. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
Assertion
Ref Expression
eldmrexrn (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem eldmrexrn
StepHypRef Expression
1 fvelrn 6897 . . 3 ((Fun 𝐹𝑌 ∈ dom 𝐹) → (𝐹𝑌) ∈ ran 𝐹)
2 eqid 2737 . . 3 (𝐹𝑌) = (𝐹𝑌)
3 eqeq1 2741 . . . 4 (𝑥 = (𝐹𝑌) → (𝑥 = (𝐹𝑌) ↔ (𝐹𝑌) = (𝐹𝑌)))
43rspcev 3537 . . 3 (((𝐹𝑌) ∈ ran 𝐹 ∧ (𝐹𝑌) = (𝐹𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
51, 2, 4sylancl 589 . 2 ((Fun 𝐹𝑌 ∈ dom 𝐹) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
65ex 416 1 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wrex 3062  dom cdm 5551  ran crn 5552  Fun wfun 6374  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388
This theorem is referenced by:  eldmrexrnb  6911
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