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Theorem hashbcval 16979
Description: Value of the "binomial set", the set of all 𝑁-element subsets of 𝐴. (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
hashbcval ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
Distinct variable groups:   𝑥,𝐶   𝑎,𝑏,𝑖,𝑥   𝐴,𝑎,𝑖,𝑥   𝑁,𝑎,𝑖,𝑥   𝑥,𝑉
Allowed substitution hints:   𝐴(𝑏)   𝐶(𝑖,𝑎,𝑏)   𝑁(𝑏)   𝑉(𝑖,𝑎,𝑏)

Proof of Theorem hashbcval
StepHypRef Expression
1 elex 3480 . 2 (𝐴𝑉𝐴 ∈ V)
2 pwexg 5378 . . . . 5 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
32adantr 479 . . . 4 ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐴 ∈ V)
4 rabexg 5334 . . . 4 (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V)
53, 4syl 17 . . 3 ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V)
6 fveqeq2 6905 . . . . . 6 (𝑏 = 𝑥 → ((♯‘𝑏) = 𝑖 ↔ (♯‘𝑥) = 𝑖))
76cbvrabv 3429 . . . . 5 {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖}
8 simpl 481 . . . . . . 7 ((𝑎 = 𝐴𝑖 = 𝑁) → 𝑎 = 𝐴)
98pweqd 4621 . . . . . 6 ((𝑎 = 𝐴𝑖 = 𝑁) → 𝒫 𝑎 = 𝒫 𝐴)
10 simpr 483 . . . . . . 7 ((𝑎 = 𝐴𝑖 = 𝑁) → 𝑖 = 𝑁)
1110eqeq2d 2736 . . . . . 6 ((𝑎 = 𝐴𝑖 = 𝑁) → ((♯‘𝑥) = 𝑖 ↔ (♯‘𝑥) = 𝑁))
129, 11rabeqbidv 3436 . . . . 5 ((𝑎 = 𝐴𝑖 = 𝑁) → {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
137, 12eqtrid 2777 . . . 4 ((𝑎 = 𝐴𝑖 = 𝑁) → {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
14 ramval.c . . . 4 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})
1513, 14ovmpoga 7575 . . 3 ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
165, 15mpd3an3 1458 . 2 ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
171, 16sylan 578 1 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {crab 3418  Vcvv 3461  𝒫 cpw 4604  cfv 6549  (class class class)co 7419  cmpo 7421  0cn0 12510  chash 14330
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 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424
This theorem is referenced by:  hashbccl  16980  hashbcss  16981  hashbc0  16982  hashbc2  16983  ramval  16985  ram0  16999  ramub1lem1  17003  ramub1lem2  17004
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