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Mirrors > Home > MPE Home > Th. List > cantnflt2 | Structured version Visualization version GIF version |
Description: An upper bound on the CNF function. (Contributed by Mario Carneiro, 28-May-2015.) (Revised by AV, 29-Jun-2019.) |
Ref | Expression |
---|---|
cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
cantnflt2.f | ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
cantnflt2.a | ⊢ (𝜑 → ∅ ∈ 𝐴) |
cantnflt2.c | ⊢ (𝜑 → 𝐶 ∈ On) |
cantnflt2.s | ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶) |
Ref | Expression |
---|---|
cantnflt2 | ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cantnfs.s | . . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵) | |
2 | cantnfs.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ On) | |
3 | cantnfs.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ On) | |
4 | eqid 2825 | . . 3 ⊢ OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅)) | |
5 | cantnflt2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
6 | eqid 2825 | . . 3 ⊢ seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·o (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +o 𝑧)), ∅) = seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·o (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +o 𝑧)), ∅) | |
7 | 1, 2, 3, 4, 5, 6 | cantnfval 8849 | . 2 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) = (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·o (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , (𝐹 supp ∅)))) |
8 | cantnflt2.a | . . 3 ⊢ (𝜑 → ∅ ∈ 𝐴) | |
9 | cantnflt2.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ On) | |
10 | cantnflt2.s | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶) | |
11 | 9, 10 | ssexd 5032 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) |
12 | 4 | oion 8717 | . . . 4 ⊢ ((𝐹 supp ∅) ∈ V → dom OrdIso( E , (𝐹 supp ∅)) ∈ On) |
13 | sucidg 6045 | . . . 4 ⊢ (dom OrdIso( E , (𝐹 supp ∅)) ∈ On → dom OrdIso( E , (𝐹 supp ∅)) ∈ suc dom OrdIso( E , (𝐹 supp ∅))) | |
14 | 11, 12, 13 | 3syl 18 | . . 3 ⊢ (𝜑 → dom OrdIso( E , (𝐹 supp ∅)) ∈ suc dom OrdIso( E , (𝐹 supp ∅))) |
15 | 1, 2, 3, 4, 5 | cantnfcl 8848 | . . . . . . 7 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom OrdIso( E , (𝐹 supp ∅)) ∈ ω)) |
16 | 15 | simpld 490 | . . . . . 6 ⊢ (𝜑 → E We (𝐹 supp ∅)) |
17 | 4 | oiiso 8718 | . . . . . 6 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → OrdIso( E , (𝐹 supp ∅)) Isom E , E (dom OrdIso( E , (𝐹 supp ∅)), (𝐹 supp ∅))) |
18 | 11, 16, 17 | syl2anc 579 | . . . . 5 ⊢ (𝜑 → OrdIso( E , (𝐹 supp ∅)) Isom E , E (dom OrdIso( E , (𝐹 supp ∅)), (𝐹 supp ∅))) |
19 | isof1o 6833 | . . . . 5 ⊢ (OrdIso( E , (𝐹 supp ∅)) Isom E , E (dom OrdIso( E , (𝐹 supp ∅)), (𝐹 supp ∅)) → OrdIso( E , (𝐹 supp ∅)):dom OrdIso( E , (𝐹 supp ∅))–1-1-onto→(𝐹 supp ∅)) | |
20 | f1ofo 6389 | . . . . 5 ⊢ (OrdIso( E , (𝐹 supp ∅)):dom OrdIso( E , (𝐹 supp ∅))–1-1-onto→(𝐹 supp ∅) → OrdIso( E , (𝐹 supp ∅)):dom OrdIso( E , (𝐹 supp ∅))–onto→(𝐹 supp ∅)) | |
21 | foima 6362 | . . . . 5 ⊢ (OrdIso( E , (𝐹 supp ∅)):dom OrdIso( E , (𝐹 supp ∅))–onto→(𝐹 supp ∅) → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) = (𝐹 supp ∅)) | |
22 | 18, 19, 20, 21 | 4syl 19 | . . . 4 ⊢ (𝜑 → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) = (𝐹 supp ∅)) |
23 | 22, 10 | eqsstrd 3864 | . . 3 ⊢ (𝜑 → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) ⊆ 𝐶) |
24 | 1, 2, 3, 4, 5, 6, 8, 14, 9, 23 | cantnflt 8853 | . 2 ⊢ (𝜑 → (seq𝜔((𝑘 ∈ V, 𝑧 ∈ V ↦ (((𝐴 ↑o (OrdIso( E , (𝐹 supp ∅))‘𝑘)) ·o (𝐹‘(OrdIso( E , (𝐹 supp ∅))‘𝑘))) +o 𝑧)), ∅)‘dom OrdIso( E , (𝐹 supp ∅))) ∈ (𝐴 ↑o 𝐶)) |
25 | 7, 24 | eqeltrd 2906 | 1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 Vcvv 3414 ⊆ wss 3798 ∅c0 4146 E cep 5256 We wwe 5304 dom cdm 5346 “ cima 5349 Oncon0 5967 suc csuc 5969 –onto→wfo 6125 –1-1-onto→wf1o 6126 ‘cfv 6127 Isom wiso 6128 (class class class)co 6910 ↦ cmpt2 6912 ωcom 7331 supp csupp 7564 seq𝜔cseqom 7813 +o coa 7828 ·o comu 7829 ↑o coe 7830 OrdIsocoi 8690 CNF ccnf 8842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-se 5306 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-isom 6136 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-supp 7565 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-seqom 7814 df-1o 7831 df-2o 7832 df-oadd 7835 df-omul 7836 df-oexp 7837 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-fsupp 8551 df-oi 8691 df-cnf 8843 |
This theorem is referenced by: cantnff 8855 cantnflem1d 8869 cnfcom3lem 8884 |
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