Theorem hashbcss 17065

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 1155	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐵 ⊆ 𝐴)
2	1	sspwd 4576	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
3		rabss2 4032	. . 3 ⊢ (𝒫 𝐵 ⊆ 𝒫 𝐴 → {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁} ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
4	2, 3	syl 18	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁} ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
5		simp1 1154	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ 𝑉)
6	5, 1	ssexd 5296	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐵 ∈ V)
7		simp3 1156	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
8		ramval.c	. . . 4 ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
9	8	hashbcval 17063	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁})
10	6, 7, 9	syl2anc 595	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁})
11	8	hashbcval 17063	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
12	11	3adant2 1149	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
13	4, 10, 12	3sstr4d 3993	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁))

Syntax hints:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ⊆ wss 3906  𝒫 cpw 4563  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  ℕ₀cn0 12505  ♯chash 14368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pow 5338  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417

This theorem is referenced by:  ramval  17069  ramub2  17075  ramub1lem2  17088