Theorem hashrabrex

Description: The number of elements in a class abstraction with a restricted existential quantification. (Contributed by Alexander van der Vekens, 29-Jul-2018)

	Ref	Expression
Hypotheses	hashrabrex.1	⊢ ( 𝜑 → 𝑌 ∈ Fin )
	hashrabrex.2	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑌 ) → { 𝑥 ∈ 𝑋 ∣ 𝜓 } ∈ Fin )
	hashrabrex.3	⊢ ( 𝜑 → Disj 𝑦 ∈ 𝑌 { 𝑥 ∈ 𝑋 ∣ 𝜓 } )
Assertion	hashrabrex	⊢ ( 𝜑 → ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ ∃ 𝑦 ∈ 𝑌 𝜓 } ) = Σ 𝑦 ∈ 𝑌 ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ 𝜓 } ) )

Proof

Step	Hyp	Ref	Expression
1		hashrabrex.1	⊢ ( 𝜑 → 𝑌 ∈ Fin )
2		hashrabrex.2	⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝑌 ) → { 𝑥 ∈ 𝑋 ∣ 𝜓 } ∈ Fin )
3		hashrabrex.3	⊢ ( 𝜑 → Disj 𝑦 ∈ 𝑌 { 𝑥 ∈ 𝑋 ∣ 𝜓 } )
4		iunrab	⊢ ∪ 𝑦 ∈ 𝑌 { 𝑥 ∈ 𝑋 ∣ 𝜓 } = { 𝑥 ∈ 𝑋 ∣ ∃ 𝑦 ∈ 𝑌 𝜓 }
5	4	eqcomi	⊢ { 𝑥 ∈ 𝑋 ∣ ∃ 𝑦 ∈ 𝑌 𝜓 } = ∪ 𝑦 ∈ 𝑌 { 𝑥 ∈ 𝑋 ∣ 𝜓 }
6	5	fveq2i	⊢ ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ ∃ 𝑦 ∈ 𝑌 𝜓 } ) = ( ♯ ‘ ∪ 𝑦 ∈ 𝑌 { 𝑥 ∈ 𝑋 ∣ 𝜓 } )
7	1 2 3	hashiun	⊢ ( 𝜑 → ( ♯ ‘ ∪ 𝑦 ∈ 𝑌 { 𝑥 ∈ 𝑋 ∣ 𝜓 } ) = Σ 𝑦 ∈ 𝑌 ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ 𝜓 } ) )
8	6 7	eqtrid	⊢ ( 𝜑 → ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ ∃ 𝑦 ∈ 𝑌 𝜓 } ) = Σ 𝑦 ∈ 𝑌 ( ♯ ‘ { 𝑥 ∈ 𝑋 ∣ 𝜓 } ) )