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Theorem eldmrexrn 7100
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 7085 . . 3 ((Fun 𝐹𝑌 ∈ dom 𝐹) → (𝐹𝑌) ∈ ran 𝐹)
2 eqid 2725 . . 3 (𝐹𝑌) = (𝐹𝑌)
3 eqeq1 2729 . . . 4 (𝑥 = (𝐹𝑌) → (𝑥 = (𝐹𝑌) ↔ (𝐹𝑌) = (𝐹𝑌)))
43rspcev 3606 . . 3 (((𝐹𝑌) ∈ ran 𝐹 ∧ (𝐹𝑌) = (𝐹𝑌)) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
51, 2, 4sylancl 584 . 2 ((Fun 𝐹𝑌 ∈ dom 𝐹) → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌))
65ex 411 1 (Fun 𝐹 → (𝑌 ∈ dom 𝐹 → ∃𝑥 ∈ ran 𝐹 𝑥 = (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  wrex 3059  dom cdm 5678  ran crn 5679  Fun wfun 6543  cfv 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-fv 6557
This theorem is referenced by:  eldmrexrnb  7101
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