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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 2823 | . . 3 ⊢ OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅)) | |
5 | cantnflt2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
6 | eqid 2823 | . . 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 9133 | . 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 7193 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
10 | 4 | oion 9002 | . . . 4 ⊢ ((𝐹 supp ∅) ∈ V → dom OrdIso( E , (𝐹 supp ∅)) ∈ On) |
11 | sucidg 6271 | . . . 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 9132 | . . . . . . 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 9003 | . . . . . 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 7078 | . . . . 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 6624 | . . . . 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 6597 | . . . . 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 4007 | . . 3 ⊢ (𝜑 → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) ⊆ 𝐶) |
24 | 1, 2, 3, 4, 5, 6, 8, 12, 13, 23 | cantnflt 9137 | . 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 2915 | 1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 ∅c0 4293 E cep 5466 We wwe 5515 dom cdm 5557 “ cima 5560 Oncon0 6193 suc csuc 6195 –onto→wfo 6355 –1-1-onto→wf1o 6356 ‘cfv 6357 Isom wiso 6358 (class class class)co 7158 ∈ cmpo 7160 ωcom 7582 supp csupp 7832 seqωcseqom 8085 +o coa 8101 ·o comu 8102 ↑o coe 8103 OrdIsocoi 8975 CNF ccnf 9126 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seqom 8086 df-1o 8104 df-2o 8105 df-oadd 8108 df-omul 8109 df-oexp 8110 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-oi 8976 df-cnf 9127 |
This theorem is referenced by: cantnff 9139 cantnflem1d 9153 cnfcom3lem 9168 |
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