Theorem fnimasnd 7362

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 6953	. . 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 1570   ∈ wcel 2145  {csn 4584   “ cima 5651   Fn wfn 6523  ‘cfv 6528

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536

This theorem is used by:  bdayn0p1  28674  esplyfval0  34115  vieta  34131