Theorem hashbcval 15706

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 3212	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		pwexg 4850	. . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
3	2	adantr 481	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐴 ∈ V)
4		rabexg 4812	. . . 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 6191	. . . . . . 7 ⊢ (𝑏 = 𝑥 → (#‘𝑏) = (#‘𝑥))
7	6	eqeq1d 2624	. . . . . 6 ⊢ (𝑏 = 𝑥 → ((#‘𝑏) = 𝑖 ↔ (#‘𝑥) = 𝑖))
8	7	cbvrabv 3199	. . . . 5 ⊢ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝑎 ∣ (#‘𝑥) = 𝑖}
9		simpl 473	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑎 = 𝐴)
10	9	pweqd 4163	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝒫 𝑎 = 𝒫 𝐴)
11		simpr 477	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑖 = 𝑁)
12	11	eqeq2d 2632	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → ((#‘𝑥) = 𝑖 ↔ (#‘𝑥) = 𝑁))
13	10, 12	rabeqbidv 3195	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → {𝑥 ∈ 𝒫 𝑎 ∣ (#‘𝑥) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
14	8, 13	syl5eq 2668	. . . 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 6790	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁} ∈ V) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
17	5, 16	mpd3an3 1425	. 2 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
18	1, 17	sylan 488	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {crab 2916  Vcvv 3200  𝒫 cpw 4158  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  ℕ₀cn0 11292  #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  hashbccl  15707  hashbcss  15708  hashbc0  15709  hashbc2  15710  ramval  15712  ram0  15726  ramub1lem1  15730  ramub1lem2  15731