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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 2737 | . . 3 ⊢ OrdIso( E , (𝐹 supp ∅)) = OrdIso( E , (𝐹 supp ∅)) | |
| 5 | cantnflt2.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
| 6 | eqid 2737 | . . 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 9708 | . 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 7466 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
| 10 | 4 | oion 9576 | . . . 4 ⊢ ((𝐹 supp ∅) ∈ V → dom OrdIso( E , (𝐹 supp ∅)) ∈ On) |
| 11 | sucidg 6465 | . . . 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 9707 | . . . . . . 7 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom OrdIso( E , (𝐹 supp ∅)) ∈ ω)) |
| 15 | 14 | simpld 494 | . . . . . 6 ⊢ (𝜑 → E We (𝐹 supp ∅)) |
| 16 | 4 | oiiso 9577 | . . . . . 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 584 | . . . . 5 ⊢ (𝜑 → OrdIso( E , (𝐹 supp ∅)) Isom E , E (dom OrdIso( E , (𝐹 supp ∅)), (𝐹 supp ∅))) |
| 18 | isof1o 7343 | . . . . 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 6855 | . . . . 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 6825 | . . . . 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 4018 | . . 3 ⊢ (𝜑 → (OrdIso( E , (𝐹 supp ∅)) “ dom OrdIso( E , (𝐹 supp ∅))) ⊆ 𝐶) |
| 24 | 1, 2, 3, 4, 5, 6, 8, 12, 13, 23 | cantnflt 9712 | . 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 2841 | 1 ⊢ (𝜑 → ((𝐴 CNF 𝐵)‘𝐹) ∈ (𝐴 ↑o 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ∅c0 4333 E cep 5583 We wwe 5636 dom cdm 5685 “ cima 5688 Oncon0 6384 suc csuc 6386 –onto→wfo 6559 –1-1-onto→wf1o 6560 ‘cfv 6561 Isom wiso 6562 (class class class)co 7431 ∈ cmpo 7433 ωcom 7887 supp csupp 8185 seqωcseqom 8487 +o coa 8503 ·o comu 8504 ↑o coe 8505 OrdIsocoi 9549 CNF ccnf 9701 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seqom 8488 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-oexp 8512 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-oi 9550 df-cnf 9702 |
| This theorem is referenced by: cantnff 9714 cantnflem1d 9728 cnfcom3lem 9743 cantnfresb 43337 |
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