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Theorem fnimasnd 38672
 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
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fnimasnd.1 . . . 4 (𝜑𝐹 Fn 𝐴)
2 fnimasnd.2 . . . 4 (𝜑𝑆𝐴)
3 fneu 6338 . . . 4 ((𝐹 Fn 𝐴𝑆𝐴) → ∃!𝑥 𝑆𝐹𝑥)
41, 2, 3syl2anc 584 . . 3 (𝜑 → ∃!𝑥 𝑆𝐹𝑥)
5 sniota 6223 . . 3 (∃!𝑥 𝑆𝐹𝑥 → {𝑥𝑆𝐹𝑥} = {(℩𝑥𝑆𝐹𝑥)})
64, 5syl 17 . 2 (𝜑 → {𝑥𝑆𝐹𝑥} = {(℩𝑥𝑆𝐹𝑥)})
7 fnrel 6331 . . . 4 (𝐹 Fn 𝐴 → Rel 𝐹)
81, 7syl 17 . . 3 (𝜑 → Rel 𝐹)
9 relimasn 5835 . . 3 (Rel 𝐹 → (𝐹 “ {𝑆}) = {𝑥𝑆𝐹𝑥})
108, 9syl 17 . 2 (𝜑 → (𝐹 “ {𝑆}) = {𝑥𝑆𝐹𝑥})
11 df-fv 6240 . . . 4 (𝐹𝑆) = (℩𝑥𝑆𝐹𝑥)
1211sneqi 4489 . . 3 {(𝐹𝑆)} = {(℩𝑥𝑆𝐹𝑥)}
1312a1i 11 . 2 (𝜑 → {(𝐹𝑆)} = {(℩𝑥𝑆𝐹𝑥)})
146, 10, 133eqtr4d 2843 1 (𝜑 → (𝐹 “ {𝑆}) = {(𝐹𝑆)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  ∃!weu 2613  {cab 2777  {csn 4478   class class class wbr 4968   “ cima 5453  Rel wrel 5455  ℩cio 6194   Fn wfn 6227  ‘cfv 6232 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-fv 6240 This theorem is referenced by: (None)
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