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Theorem fvrnressn 6086
Description: If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
Assertion
Ref Expression
fvrnressn

Proof of Theorem fvrnressn
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-ima 5017 . . 3
21eleq2i 2535 . 2
3 opeq1 4217 . . . . 5
43eleq1d 2526 . . . 4
54spcegv 3195 . . 3
6 fvex 5881 . . . 4
7 elimasng 5368 . . . 4
86, 7mpan2 671 . . 3
9 elrn2g 5198 . . . 4
106, 9mp1i 12 . . 3
115, 8, 103imtr4d 268 . 2
122, 11syl5bir 218 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  E.wex 1612  e.wcel 1818   cvv 3109  {csn 4029  <.cop 4035  rancrn 5005  |`cres 5006  "cima 5007  `cfv 5593
This theorem is referenced by:  fvn0fvelrn  6088
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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  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-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-xp 5010  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fv 5601
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