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| Mirrors > Home > MPE Home > Th. List > cantnfcl | Structured version Visualization version GIF version | ||
| Description: Basic properties of the order isomorphism 𝐺 used later. The support of an 𝐹 ∈ 𝑆 is a finite subset of 𝐴, so it is well-ordered by E and the order isomorphism has domain a finite ordinal. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
| Ref | Expression |
|---|---|
| cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
| cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
| cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
| cantnfcl.g | ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) |
| cantnfcl.f | ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
| Ref | Expression |
|---|---|
| cantnfcl | ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | suppssdm 8202 | . . . . 5 ⊢ (𝐹 supp ∅) ⊆ dom 𝐹 | |
| 2 | cantnfcl.f | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
| 3 | cantnfs.s | . . . . . . . 8 ⊢ 𝑆 = dom (𝐴 CNF 𝐵) | |
| 4 | cantnfs.a | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ On) | |
| 5 | cantnfs.b | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ On) | |
| 6 | 3, 4, 5 | cantnfs 9706 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) |
| 7 | 2, 6 | mpbid 232 | . . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)) |
| 8 | 7 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
| 9 | 1, 8 | fssdm 6755 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵) |
| 10 | onss 7805 | . . . . 5 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On) | |
| 11 | 5, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ On) |
| 12 | 9, 11 | sstrd 3994 | . . 3 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On) |
| 13 | epweon 7795 | . . 3 ⊢ E We On | |
| 14 | wess 5671 | . . 3 ⊢ ((𝐹 supp ∅) ⊆ On → ( E We On → E We (𝐹 supp ∅))) | |
| 15 | 12, 13, 14 | mpisyl 21 | . 2 ⊢ (𝜑 → E We (𝐹 supp ∅)) |
| 16 | ovexd 7466 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
| 17 | cantnfcl.g | . . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) | |
| 18 | 17 | oion 9576 | . . . . 5 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On) |
| 19 | 16, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐺 ∈ On) |
| 20 | 7 | simprd 495 | . . . . . 6 ⊢ (𝜑 → 𝐹 finSupp ∅) |
| 21 | 20 | fsuppimpd 9409 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin) |
| 22 | 17 | oien 9578 | . . . . . 6 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → dom 𝐺 ≈ (𝐹 supp ∅)) |
| 23 | 16, 15, 22 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → dom 𝐺 ≈ (𝐹 supp ∅)) |
| 24 | enfii 9226 | . . . . 5 ⊢ (((𝐹 supp ∅) ∈ Fin ∧ dom 𝐺 ≈ (𝐹 supp ∅)) → dom 𝐺 ∈ Fin) | |
| 25 | 21, 23, 24 | syl2anc 584 | . . . 4 ⊢ (𝜑 → dom 𝐺 ∈ Fin) |
| 26 | 19, 25 | elind 4200 | . . 3 ⊢ (𝜑 → dom 𝐺 ∈ (On ∩ Fin)) |
| 27 | onfin2 9268 | . . 3 ⊢ ω = (On ∩ Fin) | |
| 28 | 26, 27 | eleqtrrdi 2852 | . 2 ⊢ (𝜑 → dom 𝐺 ∈ ω) |
| 29 | 15, 28 | jca 511 | 1 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 class class class wbr 5143 E cep 5583 We wwe 5636 dom cdm 5685 Oncon0 6384 ⟶wf 6557 (class class class)co 7431 ωcom 7887 supp csupp 8185 ≈ cen 8982 Fincfn 8985 finSupp cfsupp 9401 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-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-seqom 8488 df-1o 8506 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: cantnfval2 9709 cantnfle 9711 cantnflt 9712 cantnflt2 9713 cantnff 9714 cantnfp1lem2 9719 cantnfp1lem3 9720 cantnflem1b 9726 cantnflem1d 9728 cantnflem1 9729 cnfcomlem 9739 cnfcom 9740 cnfcom2lem 9741 cnfcom3lem 9743 |
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