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Theorem eldmrexrn 5529
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 5519 . . 3 ((Fun 𝐹𝑌 ∈ dom 𝐹) → (𝐹𝑌) ∈ ran 𝐹)
2 eqid 2117 . . 3 (𝐹𝑌) = (𝐹𝑌)
3 eqeq1 2124 . . . 4 (𝑥 = (𝐹𝑌) → (𝑥 = (𝐹𝑌) ↔ (𝐹𝑌) = (𝐹𝑌)))
43rspcev 2763 . . 3 (((𝐹𝑌) ∈ ran 𝐹 ∧ (𝐹𝑌) = (𝐹𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
51, 2, 4sylancl 409 . 2 ((Fun 𝐹𝑌 ∈ dom 𝐹) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
65ex 114 1 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1316  wcel 1465  wrex 2394  dom cdm 4509  ran crn 4510  Fun wfun 5087  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by: (None)
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