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Metamath Proof Explorer

Theorem elrnmptf

Description: The range of a function in maps-to notation. Same as elrnmpt , but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypotheses	elrnmptf.1	⊢ Ⅎ 𝑥 𝐶
	elrnmptf.2	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
Assertion	elrnmptf	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elrnmptf.1	⊢ Ⅎ 𝑥 𝐶
2		elrnmptf.2	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
3	1	nfeq2	⊢ Ⅎ 𝑥 𝑦 = 𝐶
4		eqeq1	⊢ ( 𝑦 = 𝐶 → ( 𝑦 = 𝐵 ↔ 𝐶 = 𝐵 ) )
5	3 4	rexbid	⊢ ( 𝑦 = 𝐶 → ( ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )
6	2	rnmpt	⊢ ran 𝐹 = { 𝑦 ∣ ∃ 𝑥 ∈ 𝐴 𝑦 = 𝐵 }
7	5 6	elab2g	⊢ ( 𝐶 ∈ 𝑉 → ( 𝐶 ∈ ran 𝐹 ↔ ∃ 𝑥 ∈ 𝐴 𝐶 = 𝐵 ) )