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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 3185 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | pwexg 4776 | . . . . 5 ⊢ (𝐴 ∈ V → 𝒫 𝐴 ∈ V) | |
3 | 2 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → 𝒫 𝐴 ∈ V) |
4 | rabexg 4739 | . . . 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 | fveq2 6103 | . . . . . . 7 ⊢ (𝑏 = 𝑥 → (#‘𝑏) = (#‘𝑥)) | |
7 | 6 | eqeq1d 2612 | . . . . . 6 ⊢ (𝑏 = 𝑥 → ((#‘𝑏) = 𝑖 ↔ (#‘𝑥) = 𝑖)) |
8 | 7 | cbvrabv 3172 | . . . . 5 ⊢ {𝑏 ∈ 𝒫 𝑎 ∣ (#‘𝑏) = 𝑖} = {𝑥 ∈ 𝒫 𝑎 ∣ (#‘𝑥) = 𝑖} |
9 | simpl 472 | . . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑎 = 𝐴) | |
10 | 9 | pweqd 4113 | . . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝒫 𝑎 = 𝒫 𝐴) |
11 | simpr 476 | . . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → 𝑖 = 𝑁) | |
12 | 11 | eqeq2d 2620 | . . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → ((#‘𝑥) = 𝑖 ↔ (#‘𝑥) = 𝑁)) |
13 | 10, 12 | rabeqbidv 3168 | . . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 𝑁) → {𝑥 ∈ 𝒫 𝑎 ∣ (#‘𝑥) = 𝑖} = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁}) |
14 | 8, 13 | syl5eq 2656 | . . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑖 = 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}) | |
16 | 14, 15 | ovmpt2ga 6688 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0 ∧ {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁} ∈ V) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁}) |
17 | 5, 16 | mpd3an3 1417 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁}) |
18 | 1, 17 | sylan 487 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝐴𝐶𝑁) = {𝑥 ∈ 𝒫 𝐴 ∣ (#‘𝑥) = 𝑁}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 𝒫 cpw 4108 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 ℕ0cn0 11169 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 |
This theorem is referenced by: hashbccl 15545 hashbcss 15546 hashbc0 15547 hashbc2 15548 ramval 15550 ram0 15564 ramub1lem1 15568 ramub1lem2 15569 |
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