Theorem hashbcval 16940

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 3492	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		pwexg 5376	. . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
3	2	adantr 480	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐴 ∈ V)
4		rabexg 5331	. . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V)
5	3, 4	syl 17	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V)
6		fveqeq2 6900	. . . . . 6 ⊢ (𝑏 = 𝑥 → ((♯‘𝑏) = 𝑖 ↔ (♯‘𝑥) = 𝑖))
7	6	cbvrabv 3441	. . . . 5 ⊢ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖}
8		simpl 482	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑎 = 𝐴)
9	8	pweqd 4619	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝒫 𝑎 = 𝒫 𝐴)
10		simpr 484	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑖 = 𝑁)
11	10	eqeq2d 2742	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → ((♯‘𝑥) = 𝑖 ↔ (♯‘𝑥) = 𝑁))
12	9, 11	rabeqbidv 3448	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
13	7, 12	eqtrid 2783	. . . 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 7565	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
16	5, 15	mpd3an3 1461	. 2 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
17	1, 16	sylan 579	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {crab 3431  Vcvv 3473  𝒫 cpw 4602  ‘cfv 6543  (class class class)co 7412   ∈ cmpo 7414  ℕ₀cn0 12477  ♯chash 14295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417

This theorem is referenced by:  hashbccl  16941  hashbcss  16942  hashbc0  16943  hashbc2  16944  ramval  16946  ram0  16960  ramub1lem1  16964  ramub1lem2  16965