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Mirrors > Home > MPE Home > Th. List > hashsslei | Structured version Visualization version GIF version |
Description: Get an upper bound on a concretely specified finite set. Transfer boundedness to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
hashsslei.b | ⊢ 𝐵 ⊆ 𝐴 |
hashsslei.a | ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝑁) |
hashsslei.n | ⊢ 𝑁 ∈ ℕ0 |
Ref | Expression |
---|---|
hashsslei | ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashsslei.a | . . . 4 ⊢ (𝐴 ∈ Fin ∧ (♯‘𝐴) ≤ 𝑁) | |
2 | 1 | simpli 485 | . . 3 ⊢ 𝐴 ∈ Fin |
3 | hashsslei.b | . . 3 ⊢ 𝐵 ⊆ 𝐴 | |
4 | ssfi 9173 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | |
5 | 2, 3, 4 | mp2an 691 | . 2 ⊢ 𝐵 ∈ Fin |
6 | ssdomg 8996 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
7 | 2, 3, 6 | mp2 9 | . . . 4 ⊢ 𝐵 ≼ 𝐴 |
8 | hashdom 14339 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘𝐵) ≤ (♯‘𝐴) ↔ 𝐵 ≼ 𝐴)) | |
9 | 5, 2, 8 | mp2an 691 | . . . 4 ⊢ ((♯‘𝐵) ≤ (♯‘𝐴) ↔ 𝐵 ≼ 𝐴) |
10 | 7, 9 | mpbir 230 | . . 3 ⊢ (♯‘𝐵) ≤ (♯‘𝐴) |
11 | 1 | simpri 487 | . . 3 ⊢ (♯‘𝐴) ≤ 𝑁 |
12 | hashcl 14316 | . . . . . 6 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
13 | 5, 12 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐵) ∈ ℕ0 |
14 | 13 | nn0rei 12483 | . . . 4 ⊢ (♯‘𝐵) ∈ ℝ |
15 | hashcl 14316 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
16 | 2, 15 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 12483 | . . . 4 ⊢ (♯‘𝐴) ∈ ℝ |
18 | hashsslei.n | . . . . 5 ⊢ 𝑁 ∈ ℕ0 | |
19 | 18 | nn0rei 12483 | . . . 4 ⊢ 𝑁 ∈ ℝ |
20 | 14, 17, 19 | letri 11343 | . . 3 ⊢ (((♯‘𝐵) ≤ (♯‘𝐴) ∧ (♯‘𝐴) ≤ 𝑁) → (♯‘𝐵) ≤ 𝑁) |
21 | 10, 11, 20 | mp2an 691 | . 2 ⊢ (♯‘𝐵) ≤ 𝑁 |
22 | 5, 21 | pm3.2i 472 | 1 ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3949 class class class wbr 5149 ‘cfv 6544 ≼ cdom 8937 Fincfn 8939 ≤ cle 11249 ℕ0cn0 12472 ♯chash 14290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-oadd 8470 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-xnn0 12545 df-z 12559 df-uz 12823 df-fz 13485 df-hash 14291 |
This theorem is referenced by: kur14lem9 34205 |
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