Theorem fundcmpsurinjlem2

Description: Lemma 2 for fundcmpsurinj . (Contributed by AV, 4-Mar-2024)

	Ref	Expression
Hypotheses	fundcmpsurinj.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
	fundcmpsurinj.g	⊢ 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) )
Assertion	fundcmpsurinjlem2	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐺 : 𝐴 –onto→ 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		fundcmpsurinj.p	⊢ 𝑃 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
2		fundcmpsurinj.g	⊢ 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) )
3		fnex	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐹 ∈ V )
4		cnvexg	⊢ ( 𝐹 ∈ V → ◡ 𝐹 ∈ V )
5		imaexg	⊢ ( ◡ 𝐹 ∈ V → ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ∈ V )
6	3 4 5	3syl	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) → ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ∈ V )
7	6	ralrimivw	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) → ∀ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ∈ V )
8	2	fnmpt	⊢ ( ∀ 𝑥 ∈ 𝐴 ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) ∈ V → 𝐺 Fn 𝐴 )
9	7 8	syl	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐺 Fn 𝐴 )
10	1 2	fundcmpsurinjlem1	⊢ ran 𝐺 = 𝑃
11		df-fo	⊢ ( 𝐺 : 𝐴 –onto→ 𝑃 ↔ ( 𝐺 Fn 𝐴 ∧ ran 𝐺 = 𝑃 ) )
12	9 10 11	sylanblrc	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉 ) → 𝐺 : 𝐴 –onto→ 𝑃 )