Theorem hashbcval 15912

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 3361	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		pwexg 4978	. . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
3	2	adantr 466	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐴 ∈ V)
4		rabexg 4942	. . . 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		fveq2 6332	. . . . . . 7 ⊢ (𝑏 = 𝑥 → (♯‘𝑏) = (♯‘𝑥))
7	6	eqeq1d 2772	. . . . . 6 ⊢ (𝑏 = 𝑥 → ((♯‘𝑏) = 𝑖 ↔ (♯‘𝑥) = 𝑖))
8	7	cbvrabv 3348	. . . . 5 ⊢ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖}
9		simpl 468	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑎 = 𝐴)
10	9	pweqd 4300	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝒫 𝑎 = 𝒫 𝐴)
11		simpr 471	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑖 = 𝑁)
12	11	eqeq2d 2780	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → ((♯‘𝑥) = 𝑖 ↔ (♯‘𝑥) = 𝑁))
13	10, 12	rabeqbidv 3344	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
14	8, 13	syl5eq 2816	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
15		ramval.c	. . . 4 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
16	14, 15	ovmpt2ga 6936	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
17	5, 16	mpd3an3 1572	. 2 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
18	1, 17	sylan 561	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630   ∈ wcel 2144  {crab 3064  Vcvv 3349  𝒫 cpw 4295  ‘cfv 6031  (class class class)co 6792   ↦ cmpt2 6794  ℕ₀cn0 11493  ♯chash 13320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797

This theorem is referenced by:  hashbccl  15913  hashbcss  15914  hashbc0  15915  hashbc2  15916  ramval  15918  ram0  15932  ramub1lem1  15936  ramub1lem2  15937