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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 8218 | . . . . 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 9735 | . . . . . . 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 6766 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵) |
10 | onss 7820 | . . . . 5 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On) | |
11 | 5, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ On) |
12 | 9, 11 | sstrd 4019 | . . 3 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On) |
13 | epweon 7810 | . . 3 ⊢ E We On | |
14 | wess 5686 | . . 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 7483 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
17 | cantnfcl.g | . . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) | |
18 | 17 | oion 9605 | . . . . 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 9439 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin) |
22 | 17 | oien 9607 | . . . . . 6 ⊢ (((𝐹 supp ∅) ∈ V ∧ E We (𝐹 supp ∅)) → dom 𝐺 ≈ (𝐹 supp ∅)) |
23 | 16, 15, 22 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → dom 𝐺 ≈ (𝐹 supp ∅)) |
24 | enfii 9252 | . . . . 5 ⊢ (((𝐹 supp ∅) ∈ Fin ∧ dom 𝐺 ≈ (𝐹 supp ∅)) → dom 𝐺 ∈ Fin) | |
25 | 21, 23, 24 | syl2anc 583 | . . . 4 ⊢ (𝜑 → dom 𝐺 ∈ Fin) |
26 | 19, 25 | elind 4223 | . . 3 ⊢ (𝜑 → dom 𝐺 ∈ (On ∩ Fin)) |
27 | onfin2 9294 | . . 3 ⊢ ω = (On ∩ Fin) | |
28 | 26, 27 | eleqtrrdi 2855 | . 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 1537 ∈ wcel 2108 Vcvv 3488 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 class class class wbr 5166 E cep 5598 We wwe 5651 dom cdm 5700 Oncon0 6395 ⟶wf 6569 (class class class)co 7448 ωcom 7903 supp csupp 8201 ≈ cen 9000 Fincfn 9003 finSupp cfsupp 9431 OrdIsocoi 9578 CNF ccnf 9730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-seqom 8504 df-1o 8522 df-map 8886 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-oi 9579 df-cnf 9731 |
This theorem is referenced by: cantnfval2 9738 cantnfle 9740 cantnflt 9741 cantnflt2 9742 cantnff 9743 cantnfp1lem2 9748 cantnfp1lem3 9749 cantnflem1b 9755 cantnflem1d 9757 cantnflem1 9758 cnfcomlem 9768 cnfcom 9769 cnfcom2lem 9770 cnfcom3lem 9772 |
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