Theorem hashbcval 17087

Description: Value of the "binomial set", the set of all 𝑁-element subsets of 𝐴. (Contributed by Mario Carneiro, 20-Apr-2015.)

Hypothesis

Ref	Expression
ramval.c	⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})

Assertion

Ref	Expression
hashbcval	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})

Distinct variable groups:   𝑥,𝐶   𝑎,𝑏,𝑖,𝑥   𝐴,𝑎,𝑖,𝑥   𝑁,𝑎,𝑖,𝑥   𝑥,𝑉

Allowed substitution hints:   𝐴(𝑏)   𝐶(𝑖, 𝑎, 𝑏)   𝑁(𝑏)   𝑉(𝑖, 𝑎, 𝑏)

Proof of Theorem hashbcval

Step	Hyp	Ref	Expression
1		elex 3479	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		pwexg 5354	. . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
3	2	adantr 486	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐴 ∈ V)
4		rabexg 5313	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V)
5	3, 4	syl 18	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V)
6		fveqeq2 6897	. . . . . 6 ⊢ (𝑏 = 𝑥 → ((♯‘𝑏) = 𝑖 ↔ (♯‘𝑥) = 𝑖))
7	6	cbvrabv 3429	. . . . 5 ⊢ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖}
8		simpl 488	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑎 = 𝐴)
9	8	pweqd 4584	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝒫 𝑎 = 𝒫 𝐴)
10		simpr 490	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑖 = 𝑁)
11	10	eqeq2d 2777	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → ((♯‘𝑥) = 𝑖 ↔ (♯‘𝑥) = 𝑁))
12	9, 11	rabeqbidv 3437	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
13	7, 12	eqtrid 2813	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
14		ramval.c	. . . 4 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
15	13, 14	ovmpoga 7577	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
16	5, 15	mpd3an3 1491	. 2 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
17	1, 16	sylan 592	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  {crab 3419  Vcvv 3458  𝒫 cpw 4567  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  ℕ₀cn0 12522  ♯chash 14386

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pow 5341  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428

This theorem is used by:  hashbccl  17088  hashbcss  17089  hashbc0  17090  hashbc2  17091  ramval  17093  ram0  17107  ramub1lem1  17111  ramub1lem2  17112