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Theorem hashbcval 15649
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 3202 . 2 (𝐴𝑉𝐴 ∈ V)
2 pwexg 4820 . . . . 5 (𝐴 ∈ V → 𝒫 𝐴 ∈ V)
32adantr 481 . . . 4 ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐴 ∈ V)
4 rabexg 4782 . . . 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 fveq2 6158 . . . . . . 7 (𝑏 = 𝑥 → (#‘𝑏) = (#‘𝑥))
76eqeq1d 2623 . . . . . 6 (𝑏 = 𝑥 → ((#‘𝑏) = 𝑖 ↔ (#‘𝑥) = 𝑖))
87cbvrabv 3189 . . . . 5 {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝑎 ∣ (#‘𝑥) = 𝑖}
9 simpl 473 . . . . . . 7 ((𝑎 = 𝐴𝑖 = 𝑁) → 𝑎 = 𝐴)
109pweqd 4141 . . . . . 6 ((𝑎 = 𝐴𝑖 = 𝑁) → 𝒫 𝑎 = 𝒫 𝐴)
11 simpr 477 . . . . . . 7 ((𝑎 = 𝐴𝑖 = 𝑁) → 𝑖 = 𝑁)
1211eqeq2d 2631 . . . . . 6 ((𝑎 = 𝐴𝑖 = 𝑁) → ((#‘𝑥) = 𝑖 ↔ (#‘𝑥) = 𝑁))
1310, 12rabeqbidv 3185 . . . . 5 ((𝑎 = 𝐴𝑖 = 𝑁) → {𝑥 ∈ 𝒫 𝑎 ∣ (#‘𝑥) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
148, 13syl5eq 2667 . . . 4 ((𝑎 = 𝐴𝑖 = 𝑁) → {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
15 ramval.c . . . 4 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖})
1614, 15ovmpt2ga 6755 . . 3 ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁} ∈ V) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
175, 16mpd3an3 1422 . 2 ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
181, 17sylan 488 1 ((𝐴𝑉𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  𝒫 cpw 4136  cfv 5857  (class class class)co 6615  cmpt2 6617  0cn0 11252  #chash 13073
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620
This theorem is referenced by:  hashbccl  15650  hashbcss  15651  hashbc0  15652  hashbc2  15653  ramval  15655  ram0  15669  ramub1lem1  15673  ramub1lem2  15674
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