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Theorem elrnrexdm 5568
 Description: For any element in the range of a function there is an element in the domain of the function for which the function value is the element of the range. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
Assertion
Ref Expression
elrnrexdm (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐹   𝑥,𝑌

Proof of Theorem elrnrexdm
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2141 . . . . . 6 (𝑌 ∈ ran 𝐹𝑌 = 𝑌)
21ancli 321 . . . . 5 (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
32adantl 275 . . . 4 ((Fun 𝐹𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
4 eqeq2 2150 . . . . 5 (𝑦 = 𝑌 → (𝑌 = 𝑦𝑌 = 𝑌))
54rspcev 2794 . . . 4 ((𝑌 ∈ ran 𝐹𝑌 = 𝑌) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
63, 5syl 14 . . 3 ((Fun 𝐹𝑌 ∈ ran 𝐹) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
76ex 114 . 2 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦))
8 funfn 5162 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
9 eqeq2 2150 . . . 4 (𝑦 = (𝐹𝑥) → (𝑌 = 𝑦𝑌 = (𝐹𝑥)))
109rexrn 5566 . . 3 (𝐹 Fn dom 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
118, 10sylbi 120 . 2 (Fun 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
127, 11sylibd 148 1 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1332   ∈ wcel 1481  ∃wrex 2418  dom cdm 4548  ran crn 4549  Fun wfun 5126   Fn wfn 5127  ‘cfv 5132 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-sbc 2915  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-iota 5097  df-fun 5134  df-fn 5135  df-fv 5140 This theorem is referenced by:  cc2lem  7118  ennnfonelemrnh  11985  ennnfonelemf1  11987  exmidsbthrlem  13411
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