Theorem hashbcss 16946

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 1134	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐵 ⊆ 𝐴)
2	1	sspwd 4610	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
3		rabss2 4070	. . 3 ⊢ (𝒫 𝐵 ⊆ 𝒫 𝐴 → {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁} ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
4	2, 3	syl 17	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁} ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
5		simp1 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐴 ∈ 𝑉)
6	5, 1	ssexd 5317	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → 𝐵 ∈ V)
7		simp3 1135	. . 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 16944	. . 3 ⊢ ((𝐵 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁})
10	6, 7, 9	syl2anc 583	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁})
11	8	hashbcval 16944	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
12	11	3adant2 1128	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
13	4, 10, 12	3sstr4d 4024	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁))

Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {crab 3426  Vcvv 3468   ⊆ wss 3943  𝒫 cpw 4597  ‘cfv 6537  (class class class)co 7405   ∈ cmpo 7407  ℕ₀cn0 12476  ♯chash 14295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410

This theorem is referenced by:  ramval  16950  ramub2  16956  ramub1lem2  16969