Theorem fnimasnd 7342

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 6942	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → {(𝐹‘𝑆)} = (𝐹 “ {𝑆}))
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → {(𝐹‘𝑆)} = (𝐹 “ {𝑆}))
5	4	eqcomd 2736	1 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4591   “ cima 5643   Fn wfn 6508  ‘cfv 6513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-iota 6466  df-fun 6515  df-fn 6516  df-fv 6521

This theorem is referenced by:  bdayn0p1  28264