Theorem fundcmpsurinjlem1

Description: Lemma 1 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	fundcmpsurinjlem1	⊢ ran 𝐺 = 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	2	rnmpt	⊢ ran 𝐺 = { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = ( ◡ 𝐹 “ { ( 𝐹 ‘ 𝑥 ) } ) }
4	3 1	eqtr4i	⊢ ran 𝐺 = 𝑃