Theorem acni

Description: The property of being a choice set of length A . (Contributed by Mario Carneiro, 31-Aug-2015)

	Ref	Expression
Assertion	acni	⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
2	1	eleq2d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) )
3	2	ralbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) )
4	3	exbidv	⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ↔ ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) ) )
5		acnrcl	⊢ ( 𝑋 ∈ AC 𝐴 → 𝐴 ∈ V )
6		isacn	⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐴 ∈ V ) → ( 𝑋 ∈ AC 𝐴 ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) )
7	5 6	mpdan	⊢ ( 𝑋 ∈ AC 𝐴 → ( 𝑋 ∈ AC 𝐴 ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) ) )
8	7	ibi	⊢ ( 𝑋 ∈ AC 𝐴 → ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) )
9	8	adantr	⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝑓 ‘ 𝑥 ) )
10		pwexg	⊢ ( 𝑋 ∈ AC 𝐴 → 𝒫 𝑋 ∈ V )
11		difexg	⊢ ( 𝒫 𝑋 ∈ V → ( 𝒫 𝑋 ∖ { ∅ } ) ∈ V )
12	10 11	syl	⊢ ( 𝑋 ∈ AC 𝐴 → ( 𝒫 𝑋 ∖ { ∅ } ) ∈ V )
13	12 5	elmapd	⊢ ( 𝑋 ∈ AC 𝐴 → ( 𝐹 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) )
14	13	biimpar	⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → 𝐹 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) )
15	4 9 14	rspcdva	⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) )