Theorem isacnm 7523

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 3677	. . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝒫 𝑦 = 𝒫 𝑋)
2	1	rabeqdv 2809	. . . . . 6 ⊢ (𝑦 = 𝑋 → {𝑧 ∈ 𝒫 𝑦 ∣ ∃𝑗 𝑗 ∈ 𝑧} = {𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧})
3	2	oveq1d 6073	. . . . 5 ⊢ (𝑦 = 𝑋 → ({𝑧 ∈ 𝒫 𝑦 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴) = ({𝑧 ∈ 𝒫 𝑋 ∣ ∃𝑗 𝑗 ∈ 𝑧} ↑𝑚 𝐴))
4	3	raleqdv 2749	. . . 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 7489	. . 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 2967	. 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 2827	. . 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 2205  ∀wral 2522  {crab 2526  Vcvv 2815  𝒫 cpw 3674  ‘cfv 5357  (class class class)co 6058   ↑_𝑚 cmap 6895  AC wacn 7487

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 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061  df-acnm 7489

This theorem is referenced by:  finacn  7524  acnccim  7602