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| Mirrors > Home > MPE Home > Th. List > hashbcval | Structured version Visualization version GIF version | ||
| Description: Value of the "binomial set", the set of all 𝑁-element subsets of 𝐴. (Contributed by Mario Carneiro, 20-Apr-2015.) |
| Ref | Expression |
|---|---|
| ramval.c | ⊢ 𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖}) |
| Ref | Expression |
|---|---|
| hashbcval | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elex 3501 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
| 2 | pwexg 5378 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
| 3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐴 ∈ V) |
| 4 | rabexg 5337 | . . . 4 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V) | |
| 5 | 3, 4 | syl 17 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V) |
| 6 | fveqeq2 6915 | . . . . . 6 ⊢ (𝑏 = 𝑥 → ((♯‘𝑏) = 𝑖 ↔ (♯‘𝑥) = 𝑖)) | |
| 7 | 6 | cbvrabv 3447 | . . . . 5 ⊢ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖} |
| 8 | simpl 482 | . . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑎 = 𝐴) | |
| 9 | 8 | pweqd 4617 | . . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝒫 𝑎 = 𝒫 𝐴) |
| 10 | simpr 484 | . . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑖 = 𝑁) | |
| 11 | 10 | eqeq2d 2748 | . . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → ((♯‘𝑥) = 𝑖 ↔ (♯‘𝑥) = 𝑁)) |
| 12 | 9, 11 | rabeqbidv 3455 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → {𝑥 ∈ 𝒫 𝑎 ∣ (♯‘𝑥) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁}) |
| 13 | 7, 12 | eqtrid 2789 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 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}) | |
| 15 | 13, 14 | ovmpoga 7587 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁} ∈ V) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁}) |
| 16 | 5, 15 | mpd3an3 1464 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁}) |
| 17 | 1, 16 | sylan 580 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (♯‘𝑥) = 𝑁}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 𝒫 cpw 4600 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ℕ0cn0 12526 ♯chash 14369 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: hashbccl 17041 hashbcss 17042 hashbc0 17043 hashbc2 17044 ramval 17046 ram0 17060 ramub1lem1 17064 ramub1lem2 17065 |
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