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Theorem hashbcss 15632
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 ((𝐴𝑉𝐵𝐴𝑁 ∈ 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.
StepHypRef Expression
1 simp2 1060 . . . 4 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝐵𝐴)
2 sspwb 4878 . . . 4 (𝐵𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
31, 2sylib 208 . . 3 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4 rabss2 3664 . . 3 (𝒫 𝐵 ⊆ 𝒫 𝐴 → {𝑥 ∈ 𝒫 𝐵 ∣ (#‘𝑥) = 𝑁} ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
53, 4syl 17 . 2 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → {𝑥 ∈ 𝒫 𝐵 ∣ (#‘𝑥) = 𝑁} ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
6 simp1 1059 . . . 4 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝐴𝑉)
76, 1ssexd 4765 . . 3 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝐵 ∈ V)
8 simp3 1061 . . 3 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
9 ramval.c . . . 4 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
109hashbcval 15630 . . 3 ((𝐵 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (#‘𝑥) = 𝑁})
117, 8, 10syl2anc 692 . 2 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (#‘𝑥) = 𝑁})
129hashbcval 15630 . . 3 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
13123adant2 1078 . 2 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
145, 11, 133sstr4d 3627 1 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  {crab 2911  Vcvv 3186  wss 3555  𝒫 cpw 4130  cfv 5847  (class class class)co 6604  cmpt2 6606  0cn0 11236  #chash 13057
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609
This theorem is referenced by:  ramval  15636  ramub2  15642  ramub1lem2  15655
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