Theorem hashbcss 16041

Description: Subset relation for the binomial set. (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
hashbcss	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁))

Distinct variable groups:   𝑎,𝑏,𝑖   𝐴,𝑎,𝑖   𝐵,𝑎,𝑖   𝑁,𝑎,𝑖

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

Proof of Theorem hashbcss

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1168	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐵 ⊆ 𝐴)
2		sspwb 5108	. . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
3	1, 2	sylib 210	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4		rabss2 3881	. . 3 ⊢ (𝒫 𝐵 ⊆ 𝒫 𝐴 → {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁} ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
5	3, 4	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁} ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
6		simp1 1167	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ 𝑉)
7	6, 1	ssexd 5000	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐵 ∈ V)
8		simp3 1169	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
9		ramval.c	. . . 4 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
10	9	hashbcval 16039	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁})
11	7, 8, 10	syl2anc 580	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁})
12	9	hashbcval 16039	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
13	12	3adant2 1162	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
14	5, 11, 13	3sstr4d 3844	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁))

Syntax hints:   → wi 4   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  {crab 3093  Vcvv 3385   ⊆ wss 3769  𝒫 cpw 4349  ‘cfv 6101  (class class class)co 6878   ↦ cmpt2 6880  ℕ₀cn0 11580  ♯chash 13370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883

This theorem is referenced by:  ramval  16045  ramub2  16051  ramub1lem2  16064