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Theorem eldmrexrn 6722
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 6709 . . 3 ((Fun 𝐹𝑌 ∈ dom 𝐹) → (𝐹𝑌) ∈ ran 𝐹)
2 eqid 2795 . . 3 (𝐹𝑌) = (𝐹𝑌)
3 eqeq1 2799 . . . 4 (𝑥 = (𝐹𝑌) → (𝑥 = (𝐹𝑌) ↔ (𝐹𝑌) = (𝐹𝑌)))
43rspcev 3559 . . 3 (((𝐹𝑌) ∈ ran 𝐹 ∧ (𝐹𝑌) = (𝐹𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
51, 2, 4sylancl 586 . 2 ((Fun 𝐹𝑌 ∈ dom 𝐹) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
65ex 413 1 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1522  wcel 2081  wrex 3106  dom cdm 5443  ran crn 5444  Fun wfun 6219  cfv 6225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fn 6228  df-fv 6233
This theorem is referenced by:  eldmrexrnb  6723
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