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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 486 | . . 3 ⊢ 𝐴 ∈ Fin |
3 | hashsslei.b | . . 3 ⊢ 𝐵 ⊆ 𝐴 | |
4 | ssfi 8731 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ⊆ 𝐴) → 𝐵 ∈ Fin) | |
5 | 2, 3, 4 | mp2an 690 | . 2 ⊢ 𝐵 ∈ Fin |
6 | ssdomg 8548 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐵 ⊆ 𝐴 → 𝐵 ≼ 𝐴)) | |
7 | 2, 3, 6 | mp2 9 | . . . 4 ⊢ 𝐵 ≼ 𝐴 |
8 | hashdom 13737 | . . . . 5 ⊢ ((𝐵 ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘𝐵) ≤ (♯‘𝐴) ↔ 𝐵 ≼ 𝐴)) | |
9 | 5, 2, 8 | mp2an 690 | . . . 4 ⊢ ((♯‘𝐵) ≤ (♯‘𝐴) ↔ 𝐵 ≼ 𝐴) |
10 | 7, 9 | mpbir 233 | . . 3 ⊢ (♯‘𝐵) ≤ (♯‘𝐴) |
11 | 1 | simpri 488 | . . 3 ⊢ (♯‘𝐴) ≤ 𝑁 |
12 | hashcl 13714 | . . . . . 6 ⊢ (𝐵 ∈ Fin → (♯‘𝐵) ∈ ℕ0) | |
13 | 5, 12 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐵) ∈ ℕ0 |
14 | 13 | nn0rei 11902 | . . . 4 ⊢ (♯‘𝐵) ∈ ℝ |
15 | hashcl 13714 | . . . . . 6 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
16 | 2, 15 | ax-mp 5 | . . . . 5 ⊢ (♯‘𝐴) ∈ ℕ0 |
17 | 16 | nn0rei 11902 | . . . 4 ⊢ (♯‘𝐴) ∈ ℝ |
18 | hashsslei.n | . . . . 5 ⊢ 𝑁 ∈ ℕ0 | |
19 | 18 | nn0rei 11902 | . . . 4 ⊢ 𝑁 ∈ ℝ |
20 | 14, 17, 19 | letri 10762 | . . 3 ⊢ (((♯‘𝐵) ≤ (♯‘𝐴) ∧ (♯‘𝐴) ≤ 𝑁) → (♯‘𝐵) ≤ 𝑁) |
21 | 10, 11, 20 | mp2an 690 | . 2 ⊢ (♯‘𝐵) ≤ 𝑁 |
22 | 5, 21 | pm3.2i 473 | 1 ⊢ (𝐵 ∈ Fin ∧ (♯‘𝐵) ≤ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∈ wcel 2113 ⊆ wss 3929 class class class wbr 5059 ‘cfv 6348 ≼ cdom 8500 Fincfn 8502 ≤ cle 10669 ℕ0cn0 11891 ♯chash 13687 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-card 9361 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-n0 11892 df-xnn0 11962 df-z 11976 df-uz 12238 df-fz 12890 df-hash 13688 |
This theorem is referenced by: kur14lem9 32480 |
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