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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 ( ( 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 10 difexd ⊢ ( 𝑋 ∈ AC 𝐴 → ( 𝒫 𝑋 ∖ { ∅ } ) ∈ V )
12 11 5 elmapd ⊢ ( 𝑋 ∈ AC 𝐴 → ( 𝐹 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) ↔ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) )
13 12 biimpar ⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → 𝐹 ∈ ( ( 𝒫 𝑋 ∖ { ∅ } ) ↑m 𝐴 ) )
14 4 9 13 rspcdva ⊢ ( ( 𝑋 ∈ AC 𝐴 ∧ 𝐹 : 𝐴 ⟶ ( 𝒫 𝑋 ∖ { ∅ } ) ) → ∃ 𝑔 ∀ 𝑥 ∈ 𝐴 ( 𝑔 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑥 ) )