Theorem fnimasnd 7373

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	⊢ (𝜑 → 𝐹 Fn 𝐴)
fnimasnd.2	⊢ (𝜑 → 𝑆 ∈ 𝐴)

Assertion

Ref	Expression
fnimasnd	⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)})

Proof of Theorem fnimasnd

Step	Hyp	Ref	Expression
1		fnimasnd.1	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fnimasnd.2	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
3		fnsnfv 6964	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → {(𝐹‘𝑆)} = (𝐹 “ {𝑆}))
4	1, 2, 3	syl2anc 596	. 2 ⊢ (𝜑 → {(𝐹‘𝑆)} = (𝐹 “ {𝑆}))
5	4	eqcomd 2766	1 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)})

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  {csn 4583   “ cima 5653   Fn wfn 6533  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  bdayn0p1  28755  esplyfval0  34196  vieta  34212