Theorem fnimasnd 7315

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 6915	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → {(𝐹‘𝑆)} = (𝐹 “ {𝑆}))
4	1, 2, 3	syl2anc 585	. 2 ⊢ (𝜑 → {(𝐹‘𝑆)} = (𝐹 “ {𝑆}))
5	4	eqcomd 2743	1 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4568   “ cima 5629   Fn wfn 6489  ‘cfv 6494

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-iota 6450  df-fun 6496  df-fn 6497  df-fv 6502

This theorem is referenced by:  bdayn0p1  28379  esplyfval0  33727  vieta  33743