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Theorem eldmrexrnb 6038
Description: For any element in the domain of a function, there is an element in the range of the function which is the value of the function at that element. Because of the definition df-fv 5601 of the value of a function, the theorem is only valid in general if the empty set is not contained in the range of the function (the implication "to the right" is always valid). Indeed, with the definition df-fv 5601 of the value of a function, may mean that the value of at is the empty set or that is not defined at . (Contributed by Alexander van der Vekens, 17-Dec-2017.)
Assertion
Ref Expression
eldmrexrnb
Distinct variable groups:   ,   ,

Proof of Theorem eldmrexrnb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eldmrexrn 6037 . . 3
21adantr 465 . 2
3 eleq1 2529 . . . . 5
4 elnelne2 2805 . . . . . . . . 9
5 n0 3794 . . . . . . . . . 10
6 elfvdm 5897 . . . . . . . . . . 11
76exlimiv 1722 . . . . . . . . . 10
85, 7sylbi 195 . . . . . . . . 9
94, 8syl 16 . . . . . . . 8
109expcom 435 . . . . . . 7
1110adantl 466 . . . . . 6
1211com12 31 . . . . 5
133, 12syl6bi 228 . . . 4
1413com13 80 . . 3
1514rexlimdv 2947 . 2
162, 15impbid 191 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  =/=wne 2652  e/wnel 2653  E.wrex 2808   c0 3784  domcdm 5004  rancrn 5005  Funwfun 5587  `cfv 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601
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