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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 7845 | . . . . 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 9131 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) |
7 | 2, 6 | mpbid 234 | . . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)) |
8 | 7 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
9 | 1, 8 | fssdm 6532 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵) |
10 | onss 7507 | . . . . 5 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On) | |
11 | 5, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ On) |
12 | 9, 11 | sstrd 3979 | . . 3 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On) |
13 | epweon 7499 | . . 3 ⊢ E We On | |
14 | wess 5544 | . . 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 7193 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
17 | cantnfcl.g | . . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) | |
18 | 17 | oion 9002 | . . . . 5 ⊢ ((𝐹 supp ∅) ∈ V → dom 𝐺 ∈ On) |
19 | 16, 18 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝐺 ∈ On) |
20 | 7 | simprd 498 | . . . . . 6 ⊢ (𝜑 → 𝐹 finSupp ∅) |
21 | 20 | fsuppimpd 8842 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin) |
22 | 17 | oien 9004 | . . . . . 6 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → dom 𝐺 ≈ (𝐹 supp ∅)) |
23 | 16, 15, 22 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → dom 𝐺 ≈ (𝐹 supp ∅)) |
24 | enfii 8737 | . . . . 5 ⊢ (((𝐹 supp ∅) ∈ Fin ∧ dom 𝐺 ≈ (𝐹 supp ∅)) → dom 𝐺 ∈ Fin) | |
25 | 21, 23, 24 | syl2anc 586 | . . . 4 ⊢ (𝜑 → dom 𝐺 ∈ Fin) |
26 | 19, 25 | elind 4173 | . . 3 ⊢ (𝜑 → dom 𝐺 ∈ (On ∩ Fin)) |
27 | onfin2 8712 | . . 3 ⊢ ω = (On ∩ Fin) | |
28 | 26, 27 | eleqtrrdi 2926 | . 2 ⊢ (𝜑 → dom 𝐺 ∈ ω) |
29 | 15, 28 | jca 514 | 1 ⊢ (𝜑 → ( E We (𝐹 supp ∅) ∧ dom 𝐺 ∈ ω)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 class class class wbr 5068 E cep 5466 We wwe 5515 dom cdm 5557 Oncon0 6193 ⟶wf 6353 (class class class)co 7158 ωcom 7582 supp csupp 7832 ≈ cen 8508 Fincfn 8511 finSupp cfsupp 8835 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-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-seqom 8086 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: cantnfval2 9134 cantnfle 9136 cantnflt 9137 cantnflt2 9138 cantnff 9139 cantnfp1lem2 9144 cantnfp1lem3 9145 cantnflem1b 9151 cantnflem1d 9153 cantnflem1 9154 cnfcomlem 9164 cnfcom 9165 cnfcom2lem 9166 cnfcom3lem 9168 |
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