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

Assertion

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

Proof of Theorem fnimasnd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnimasnd.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn 𝐴)
2		fnimasnd.2	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
3		fneu 6338	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝐴) → ∃!𝑥 𝑆𝐹𝑥)
4	1, 2, 3	syl2anc 584	. . 3 ⊢ (𝜑 → ∃!𝑥 𝑆𝐹𝑥)
5		sniota 6223	. . 3 ⊢ (∃!𝑥 𝑆𝐹𝑥 → {𝑥 ∣ 𝑆𝐹𝑥} = {(℩𝑥𝑆𝐹𝑥)})
6	4, 5	syl 17	. 2 ⊢ (𝜑 → {𝑥 ∣ 𝑆𝐹𝑥} = {(℩𝑥𝑆𝐹𝑥)})
7		fnrel 6331	. . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
8	1, 7	syl 17	. . 3 ⊢ (𝜑 → Rel 𝐹)
9		relimasn 5835	. . 3 ⊢ (Rel 𝐹 → (𝐹 “ {𝑆}) = {𝑥 ∣ 𝑆𝐹𝑥})
10	8, 9	syl 17	. 2 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {𝑥 ∣ 𝑆𝐹𝑥})
11		df-fv 6240	. . . 4 ⊢ (𝐹‘𝑆) = (℩𝑥𝑆𝐹𝑥)
12	11	sneqi 4489	. . 3 ⊢ {(𝐹‘𝑆)} = {(℩𝑥𝑆𝐹𝑥)}
13	12	a1i 11	. 2 ⊢ (𝜑 → {(𝐹‘𝑆)} = {(℩𝑥𝑆𝐹𝑥)})
14	6, 10, 13	3eqtr4d 2843	1 ⊢ (𝜑 → (𝐹 “ {𝑆}) = {(𝐹‘𝑆)})

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)