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Theorem elrnrexdm 6015
 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
Distinct variable groups:   ,   ,

Proof of Theorem elrnrexdm
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqidd 2405 . . . . . 6
21ancli 551 . . . . 5
32adantl 466 . . . 4
4 eqeq2 2419 . . . . 5
54rspcev 3162 . . . 4
63, 5syl 17 . . 3
76ex 434 . 2
8 funfn 5600 . . 3
9 eqeq2 2419 . . . 4
109rexrn 6013 . . 3
118, 10sylbi 197 . 2
127, 11sylibd 216 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 186   wa 369   wceq 1407   wcel 1844  wrex 2757   cdm 4825   crn 4826   wfun 5565   wfn 5566  cfv 5571 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-iota 5535  df-fun 5573  df-fn 5574  df-fv 5579 This theorem is referenced by:  nbgraf1olem1  24870  vdusgra0nedg  25337  usgravd0nedg  25347  bj-ccinftydisj  31193
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