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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 2821 | . . 3 ⊢ OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅)) | |
5 | cantnflt2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
6 | eqid 2821 | . . 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 9125 | . 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 | ovexd 7185 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
10 | 4 | oion 8994 | . . . 4 ⊢ ((𝐹 supp ∅) ∈ V → dom OrdIso( E , (𝐹 supp ∅)) ∈ On) |
11 | sucidg 6264 | . . . 4 ⊢ (dom OrdIso( E , (𝐹 supp ∅)) ∈ On → dom OrdIso( E , (𝐹 supp ∅)) ∈ suc dom OrdIso( E , (𝐹 supp ∅))) | |
12 | 9, 10, 11 | 3syl 18 | . . 3 ⊢ (𝜑 → dom OrdIso( E , (𝐹 supp ∅)) ∈ suc dom OrdIso( E , (𝐹 supp ∅))) |
13 | cantnflt2.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ On) | |
14 | 1, 2, 3, 4, 5 | cantnfcl 9124 | . . . . . . 7 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom OrdIso( E , (𝐹 supp ∅)) ∈ ω)) |
15 | 14 | simpld 497 | . . . . . 6 ⊢ (𝜑 → E We (𝐹 supp ∅)) |
16 | 4 | oiiso 8995 | . . . . . 6 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → OrdIso( E , (𝐹 supp ∅)) Isom E , E (dom OrdIso( E , (𝐹 supp ∅)), (𝐹 supp ∅))) |
17 | 9, 15, 16 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → OrdIso( E , (𝐹 supp ∅)) Isom E , E (dom OrdIso( E , (𝐹 supp ∅)), (𝐹 supp ∅))) |
18 | isof1o 7070 | . . . . 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 ∅)) | |
19 | f1ofo 6617 | . . . . 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 ∅)) | |
20 | foima 6590 | . . . . 5 ⊢ (OrdIso( E , (𝐹 supp ∅)):dom OrdIso( E , (𝐹 supp ∅))–onto→(𝐹 supp ∅) → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) = (𝐹 supp ∅)) | |
21 | 17, 18, 19, 20 | 4syl 19 | . . . 4 ⊢ (𝜑 → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) = (𝐹 supp ∅)) |
22 | cantnflt2.s | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐶) | |
23 | 21, 22 | eqsstrd 4005 | . . 3 ⊢ (𝜑 → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) ⊆ 𝐶) |
24 | 1, 2, 3, 4, 5, 6, 8, 12, 13, 23 | cantnflt 9129 | . 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 2913 | 1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ⊆ wss 3936 ∅c0 4291 E cep 5459 We wwe 5508 dom cdm 5550 “ cima 5553 Oncon0 6186 suc csuc 6188 –onto→wfo 6348 –1-1-onto→wf1o 6349 ‘cfv 6350 Isom wiso 6351 (class class class)co 7150 ∈ cmpo 7152 ωcom 7574 supp csupp 7824 seqωcseqom 8077 +o coa 8093 ·o comu 8094 ↑o coe 8095 OrdIsocoi 8967 CNF ccnf 9118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-seqom 8078 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-oexp 8102 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-oi 8968 df-cnf 9119 |
This theorem is referenced by: cantnff 9131 cantnflem1d 9145 cnfcom3lem 9160 |
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