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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 2610 | . . 3 ⊢ OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅)) | |
5 | cantnflt2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
6 | eqid 2610 | . . 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 8448 | . 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 4733 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) |
12 | 4 | oion 8324 | . . . 4 ⊢ ((𝐹 supp ∅) ∈ V → dom OrdIso( E , (𝐹 supp ∅)) ∈ On) |
13 | sucidg 5720 | . . . 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 8447 | . . . . . . 7 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom OrdIso( E , (𝐹 supp ∅)) ∈ ω)) |
16 | 15 | simpld 474 | . . . . . 6 ⊢ (𝜑 → E We (𝐹 supp ∅)) |
17 | 4 | oiiso 8325 | . . . . . 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 691 | . . . . 5 ⊢ (𝜑 → OrdIso( E , (𝐹 supp ∅)) Isom E , E (dom OrdIso( E , (𝐹 supp ∅)), (𝐹 supp ∅))) |
19 | isof1o 6473 | . . . . 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 6057 | . . . . 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 6033 | . . . . 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 3602 | . . 3 ⊢ (𝜑 → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) ⊆ 𝐶) |
24 | 1, 2, 3, 4, 5, 6, 8, 14, 9, 23 | cantnflt 8452 | . 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 2688 | 1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑𝑜 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 E cep 4947 We wwe 4996 dom cdm 5038 “ cima 5041 Oncon0 5640 suc csuc 5642 –onto→wfo 5802 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ↦ cmpt2 6551 ωcom 6957 supp csupp 7182 seq𝜔cseqom 7429 +𝑜 coa 7444 ·𝑜 comu 7445 ↑𝑜 coe 7446 OrdIsocoi 8297 CNF ccnf 8441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-oexp 7453 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-cnf 8442 |
This theorem is referenced by: cantnff 8454 cantnflem1d 8468 cnfcom3lem 8483 |
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