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Theorem elrnrexdm 5333
 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 2057 . . . . . 6 (𝑌 ∈ ran 𝐹𝑌 = 𝑌)
21ancli 310 . . . . 5 (𝑌 ∈ ran 𝐹 → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
32adantl 266 . . . 4 ((Fun 𝐹𝑌 ∈ ran 𝐹) → (𝑌 ∈ ran 𝐹𝑌 = 𝑌))
4 eqeq2 2065 . . . . 5 (𝑦 = 𝑌 → (𝑌 = 𝑦𝑌 = 𝑌))
54rspcev 2673 . . . 4 ((𝑌 ∈ ran 𝐹𝑌 = 𝑌) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
63, 5syl 14 . . 3 ((Fun 𝐹𝑌 ∈ ran 𝐹) → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦)
76ex 112 . 2 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦))
8 funfn 4958 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
9 eqeq2 2065 . . . 4 (𝑦 = (𝐹𝑥) → (𝑌 = 𝑦𝑌 = (𝐹𝑥)))
109rexrn 5331 . . 3 (𝐹 Fn dom 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
118, 10sylbi 118 . 2 (Fun 𝐹 → (∃𝑦 ∈ ran 𝐹 𝑌 = 𝑦 ↔ ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
127, 11sylibd 142 1 (Fun 𝐹 → (𝑌 ∈ ran 𝐹 → ∃𝑥 ∈ dom 𝐹 𝑌 = (𝐹𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  ∃wrex 2324  dom cdm 4372  ran crn 4373  Fun wfun 4923   Fn wfn 4924  ‘cfv 4929 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937 This theorem is referenced by: (None)
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