Metamath Proof Explorer


Theorem acni

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

Ref Expression
Assertion acni ( ( 𝑋AC 𝐴𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∃ 𝑔𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝐹𝑥 ) )

Proof

Step Hyp Ref Expression
1 fveq1 ( 𝑓 = 𝐹 → ( 𝑓𝑥 ) = ( 𝐹𝑥 ) )
2 1 eleq2d ( 𝑓 = 𝐹 → ( ( 𝑔𝑥 ) ∈ ( 𝑓𝑥 ) ↔ ( 𝑔𝑥 ) ∈ ( 𝐹𝑥 ) ) )
3 2 ralbidv ( 𝑓 = 𝐹 → ( ∀ 𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝑓𝑥 ) ↔ ∀ 𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝐹𝑥 ) ) )
4 3 exbidv ( 𝑓 = 𝐹 → ( ∃ 𝑔𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝑓𝑥 ) ↔ ∃ 𝑔𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝐹𝑥 ) ) )
5 acnrcl ( 𝑋AC 𝐴𝐴 ∈ V )
6 isacn ( ( 𝑋AC 𝐴𝐴 ∈ V ) → ( 𝑋AC 𝐴 ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝑓𝑥 ) ) )
7 5 6 mpdan ( 𝑋AC 𝐴 → ( 𝑋AC 𝐴 ↔ ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝑓𝑥 ) ) )
8 7 ibi ( 𝑋AC 𝐴 → ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝑓𝑥 ) )
9 8 adantr ( ( 𝑋AC 𝐴𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∀ 𝑓 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ∃ 𝑔𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝑓𝑥 ) )
10 pwexg ( 𝑋AC 𝐴 → 𝒫 𝑋 ∈ V )
11 10 difexd ( 𝑋AC 𝐴 → ( 𝒫 𝑋 ∖ { ∅ } ) ∈ V )
12 11 5 elmapd ( 𝑋AC 𝐴 → ( 𝐹 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) )
13 12 biimpar ( ( 𝑋AC 𝐴𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → 𝐹 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) )
14 4 9 13 rspcdva ( ( 𝑋AC 𝐴𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∃ 𝑔𝑥𝐴 ( 𝑔𝑥 ) ∈ ( 𝐹𝑥 ) )