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Theorem hashbcss 16333
 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 1131 . . . 4 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝐵𝐴)
2 sspwb 5337 . . . 4 (𝐵𝐴 ↔ 𝒫 𝐵 ⊆ 𝒫 𝐴)
31, 2sylib 219 . . 3 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝒫 𝐵 ⊆ 𝒫 𝐴)
4 rabss2 4057 . . 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 1130 . . . 4 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝐴𝑉)
76, 1ssexd 5224 . . 3 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝐵 ∈ V)
8 simp3 1132 . . 3 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → 𝑁 ∈ ℕ0)
9 ramval.c . . . 4 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
109hashbcval 16331 . . 3 ((𝐵 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁})
117, 8, 10syl2anc 584 . 2 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) = {𝑥 ∈ 𝒫 𝐵 ∣ (♯‘𝑥) = 𝑁})
129hashbcval 16331 . . 3 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
13123adant2 1125 . 2 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
145, 11, 133sstr4d 4017 1 ((𝐴𝑉𝐵𝐴𝑁 ∈ ℕ0) → (𝐵𝐶𝑁) ⊆ (𝐴𝐶𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  {crab 3146  Vcvv 3499   ⊆ wss 3939  𝒫 cpw 4541  ‘cfv 6351  (class class class)co 7151   ∈ cmpo 7153  ℕ0cn0 11889  ♯chash 13683 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-ov 7154  df-oprab 7155  df-mpo 7156 This theorem is referenced by:  ramval  16337  ramub2  16343  ramub1lem2  16356
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