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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 7897 | . . . . 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 9259 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) |
7 | 2, 6 | mpbid 235 | . . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)) |
8 | 7 | simpld 498 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
9 | 1, 8 | fssdm 6543 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵) |
10 | onss 7546 | . . . . 5 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On) | |
11 | 5, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ On) |
12 | 9, 11 | sstrd 3897 | . . 3 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On) |
13 | epweon 7538 | . . 3 ⊢ E We On | |
14 | wess 5523 | . . 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 7226 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
17 | cantnfcl.g | . . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) | |
18 | 17 | oion 9130 | . . . . 5 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On) |
19 | 16, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐺 ∈ On) |
20 | 7 | simprd 499 | . . . . . 6 ⊢ (𝜑 → 𝐹 finSupp ∅) |
21 | 20 | fsuppimpd 8970 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin) |
22 | 17 | oien 9132 | . . . . . 6 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → dom 𝐺 ≈ (𝐹 supp ∅)) |
23 | 16, 15, 22 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → dom 𝐺 ≈ (𝐹 supp ∅)) |
24 | enfii 8841 | . . . . 5 ⊢ (((𝐹 supp ∅) ∈ Fin ∧ dom 𝐺 ≈ (𝐹 supp ∅)) → dom 𝐺 ∈ Fin) | |
25 | 21, 23, 24 | syl2anc 587 | . . . 4 ⊢ (𝜑 → dom 𝐺 ∈ Fin) |
26 | 19, 25 | elind 4094 | . . 3 ⊢ (𝜑 → dom 𝐺 ∈ (On ∩ Fin)) |
27 | onfin2 8847 | . . 3 ⊢ ω = (On ∩ Fin) | |
28 | 26, 27 | eleqtrrdi 2842 | . 2 ⊢ (𝜑 → dom 𝐺 ∈ ω) |
29 | 15, 28 | jca 515 | 1 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 class class class wbr 5039 E cep 5444 We wwe 5493 dom cdm 5536 Oncon0 6191 ⟶wf 6354 (class class class)co 7191 ωcom 7622 supp csupp 7881 ≈ cen 8601 Fincfn 8604 finSupp cfsupp 8963 OrdIsocoi 9103 CNF ccnf 9254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-seqom 8162 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-oi 9104 df-cnf 9255 |
This theorem is referenced by: cantnfval2 9262 cantnfle 9264 cantnflt 9265 cantnflt2 9266 cantnff 9267 cantnfp1lem2 9272 cantnfp1lem3 9273 cantnflem1b 9279 cantnflem1d 9281 cantnflem1 9282 cnfcomlem 9292 cnfcom 9293 cnfcom2lem 9294 cnfcom3lem 9296 |
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