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Theorem eldmrexrn 5653
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 5643 . . 3 ((Fun 𝐹𝑌 ∈ dom 𝐹) → (𝐹𝑌) ∈ ran 𝐹)
2 eqid 2177 . . 3 (𝐹𝑌) = (𝐹𝑌)
3 eqeq1 2184 . . . 4 (𝑥 = (𝐹𝑌) → (𝑥 = (𝐹𝑌) ↔ (𝐹𝑌) = (𝐹𝑌)))
43rspcev 2841 . . 3 (((𝐹𝑌) ∈ ran 𝐹 ∧ (𝐹𝑌) = (𝐹𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
51, 2, 4sylancl 413 . 2 ((Fun 𝐹𝑌 ∈ dom 𝐹) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
65ex 115 1 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wrex 2456  dom cdm 4623  ran crn 4624  Fun wfun 5206  cfv 5212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220
This theorem is referenced by: (None)
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