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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 7846 | . . . . 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 9132 | . . . . . . 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 6533 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵) |
10 | onss 7508 | . . . . 5 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On) | |
11 | 5, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ On) |
12 | 9, 11 | sstrd 3980 | . . 3 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On) |
13 | epweon 7500 | . . 3 ⊢ E We On | |
14 | wess 5545 | . . 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 7194 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
17 | cantnfcl.g | . . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) | |
18 | 17 | oion 9003 | . . . . 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 8843 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin) |
22 | 17 | oien 9005 | . . . . . 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 8738 | . . . . 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 4174 | . . 3 ⊢ (𝜑 → dom 𝐺 ∈ (On ∩ Fin)) |
27 | onfin2 8713 | . . 3 ⊢ ω = (On ∩ Fin) | |
28 | 26, 27 | eleqtrrdi 2927 | . 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 1536 ∈ wcel 2113 Vcvv 3497 ∩ cin 3938 ⊆ wss 3939 ∅c0 4294 class class class wbr 5069 E cep 5467 We wwe 5516 dom cdm 5558 Oncon0 6194 ⟶wf 6354 (class class class)co 7159 ωcom 7583 supp csupp 7833 ≈ cen 8509 Fincfn 8512 finSupp cfsupp 8836 OrdIsocoi 8976 CNF ccnf 9127 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 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 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-seqom 8087 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-oi 8977 df-cnf 9128 |
This theorem is referenced by: cantnfval2 9135 cantnfle 9137 cantnflt 9138 cantnflt2 9139 cantnff 9140 cantnfp1lem2 9145 cantnfp1lem3 9146 cantnflem1b 9152 cantnflem1d 9154 cantnflem1 9155 cnfcomlem 9165 cnfcom 9166 cnfcom2lem 9167 cnfcom3lem 9169 |
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