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Theorem elrnrexdmb 5232
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, 17-Dec-2017.)
Assertion
Ref Expression
elrnrexdmb (Fun 𝐹 → (𝑌 ran 𝐹x dom 𝐹 𝑌 = (𝐹x)))
Distinct variable groups:   x,𝐹   x,𝑌

Proof of Theorem elrnrexdmb
StepHypRef Expression
1 funfn 4857 . . 3 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fvelrnb 5146 . . 3 (𝐹 Fn dom 𝐹 → (𝑌 ran 𝐹x dom 𝐹(𝐹x) = 𝑌))
31, 2sylbi 114 . 2 (Fun 𝐹 → (𝑌 ran 𝐹x dom 𝐹(𝐹x) = 𝑌))
4 eqcom 2024 . . 3 (𝑌 = (𝐹x) ↔ (𝐹x) = 𝑌)
54rexbii 2309 . 2 (x dom 𝐹 𝑌 = (𝐹x) ↔ x dom 𝐹(𝐹x) = 𝑌)
63, 5syl6bbr 187 1 (Fun 𝐹 → (𝑌 ran 𝐹x dom 𝐹 𝑌 = (𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1228   wcel 1374  wrex 2285  dom cdm 4272  ran crn 4273  Fun wfun 4823   Fn wfn 4824  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by: (None)
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