Theorem isacnm 7509

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

Assertion

Ref	Expression
isacnm	⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))

Distinct variable groups:   𝑓,𝑔,𝑥,𝑧,𝑗,𝐴   𝑓,𝑋,𝑔,𝑥,𝑧

Allowed substitution hints:   𝑉(𝑥,𝑧,𝑓,𝑔,𝑗)   𝑊(𝑥,𝑧,𝑓,𝑔,𝑗)   𝑋(𝑗)

Proof of Theorem isacnm

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 3671	. . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
2	1	rabeqdv 2806	. . . . . 6 ⊢ (𝑦 = 𝑋 → {𝑧 ∈ 𝒫 𝑦 ∣ ∃𝑗 𝑗 ∈ 𝑧} = {𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧})
3	2	oveq1d 6064	. . . . 5 ⊢ (𝑦 = 𝑋 → ({𝑧 ∈ 𝒫 𝑦 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴) = ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴))
4	3	raleqdv 2746	. . . 4 ⊢ (𝑦 = 𝑋 → (∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑦 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥) ↔ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
5	4	anbi2d 464	. . 3 ⊢ (𝑦 = 𝑋 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑦 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))))
6		df-acnm 7475	. . 3 ⊢ AC 𝐴 = {𝑦 ∣ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑦 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))}
7	5, 6	elab2g 2963	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∈ AC 𝐴 ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥))))
8		elex 2824	. . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V)
9		biid 171	. . . 4 ⊢ ((𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ (𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
10	9	baib 927	. . 3 ⊢ (𝐴 ∈ V → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
11	8, 10	syl 14	. 2 ⊢ (𝐴 ∈ 𝑊 → ((𝐴 ∈ V ∧ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)) ↔ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))
12	7, 11	sylan9bb 462	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝑋 ∈ AC 𝐴 ↔ ∀𝑓 ∈ ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴)∃𝑔∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ (𝑓‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  {crab 2524  Vcvv 2812  𝒫 cpw 3668  ‘cfv 5351  (class class class)co 6049   ↑_𝑚 cmap 6881  AC wacn 7473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-iota 5311  df-fv 5359  df-ov 6052  df-acnm 7475

This theorem is referenced by:  finacn  7510  acnccim  7582