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Theorem elrnrexdm 5249
 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 𝐹x 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 2038 . . . . . 6 (𝑌 ran 𝐹𝑌 = 𝑌)
21ancli 306 . . . . 5 (𝑌 ran 𝐹 → (𝑌 ran 𝐹 𝑌 = 𝑌))
32adantl 262 . . . 4 ((Fun 𝐹 𝑌 ran 𝐹) → (𝑌 ran 𝐹 𝑌 = 𝑌))
4 eqeq2 2046 . . . . 5 (y = 𝑌 → (𝑌 = y𝑌 = 𝑌))
54rspcev 2650 . . . 4 ((𝑌 ran 𝐹 𝑌 = 𝑌) → y ran 𝐹 𝑌 = y)
63, 5syl 14 . . 3 ((Fun 𝐹 𝑌 ran 𝐹) → y ran 𝐹 𝑌 = y)
76ex 108 . 2 (Fun 𝐹 → (𝑌 ran 𝐹y ran 𝐹 𝑌 = y))
8 funfn 4874 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
9 eqeq2 2046 . . . 4 (y = (𝐹x) → (𝑌 = y𝑌 = (𝐹x)))
109rexrn 5247 . . 3 (𝐹 Fn dom 𝐹 → (y ran 𝐹 𝑌 = yx dom 𝐹 𝑌 = (𝐹x)))
118, 10sylbi 114 . 2 (Fun 𝐹 → (y ran 𝐹 𝑌 = yx dom 𝐹 𝑌 = (𝐹x)))
127, 11sylibd 138 1 (Fun 𝐹 → (𝑌 ran 𝐹x dom 𝐹 𝑌 = (𝐹x)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853 This theorem is referenced by: (None)
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