Theorem elsetpreimafvssdm

Description: An element of the class P of all preimages of function values is a subset of the domain of the function. (Contributed by AV, 8-Mar-2024)

	Ref	Expression
Hypothesis	setpreimafvex.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
Assertion	elsetpreimafvssdm	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ) → 𝑆 ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		setpreimafvex.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
2	1	elsetpreimafv	⊢ ( 𝑆 ∈ 𝑃 → ∃ 𝑥 ∈ 𝐴 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) )
3		cnvimass	⊢ ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ⊆ dom 𝐹
4		fndm	⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 )
5	3 4	sseqtrid	⊢ ( 𝐹 Fn 𝐴 → ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ⊆ 𝐴 )
6	5	adantr	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ⊆ 𝐴 )
7		sseq1	⊢ ( 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) → ( 𝑆 ⊆ 𝐴 ↔ ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ⊆ 𝐴 ) )
8	6 7	syl5ibrcom	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) → 𝑆 ⊆ 𝐴 ) )
9	8	expcom	⊢ ( 𝑥 ∈ 𝐴 → ( 𝐹 Fn 𝐴 → ( 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) → 𝑆 ⊆ 𝐴 ) ) )
10	9	com23	⊢ ( 𝑥 ∈ 𝐴 → ( 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) → ( 𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴 ) ) )
11	10	rexlimiv	⊢ ( ∃ 𝑥 ∈ 𝐴 𝑆 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) → ( 𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴 ) )
12	2 11	syl	⊢ ( 𝑆 ∈ 𝑃 → ( 𝐹 Fn 𝐴 → 𝑆 ⊆ 𝐴 ) )
13	12	impcom	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑆 ∈ 𝑃 ) → 𝑆 ⊆ 𝐴 )