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Theorem fnimasnd 39715
 Description: The image of a function by a singleton whose element is in the domain of the function. (Contributed by Steven Nguyen, 7-Jun-2023.)
Hypotheses
Ref Expression
fnimasnd.1 (𝜑𝐹 Fn 𝐴)
fnimasnd.2 (𝜑𝑆𝐴)
Assertion
Ref Expression
fnimasnd (𝜑 → (𝐹 “ {𝑆}) = {(𝐹𝑆)})

Proof of Theorem fnimasnd
StepHypRef Expression
1 fnimasnd.1 . . 3 (𝜑𝐹 Fn 𝐴)
2 fnimasnd.2 . . 3 (𝜑𝑆𝐴)
3 fnsnfv 6731 . . 3 ((𝐹 Fn 𝐴𝑆𝐴) → {(𝐹𝑆)} = (𝐹 “ {𝑆}))
41, 2, 3syl2anc 587 . 2 (𝜑 → {(𝐹𝑆)} = (𝐹 “ {𝑆}))
54eqcomd 2764 1 (𝜑 → (𝐹 “ {𝑆}) = {(𝐹𝑆)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  {csn 4522   “ cima 5527   Fn wfn 6330  ‘cfv 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-fv 6343 This theorem is referenced by: (None)
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