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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 8159 | . . . . 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 9660 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) |
7 | 2, 6 | mpbid 231 | . . . . . 6 ⊢ (𝜑 → (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅)) |
8 | 7 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝐹:𝐵⟶𝐴) |
9 | 1, 8 | fssdm 6730 | . . . 4 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ 𝐵) |
10 | onss 7768 | . . . . 5 ⊢ (𝐵 ∈ On → 𝐵 ⊆ On) | |
11 | 5, 10 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐵 ⊆ On) |
12 | 9, 11 | sstrd 3987 | . . 3 ⊢ (𝜑 → (𝐹 supp ∅) ⊆ On) |
13 | epweon 7758 | . . 3 ⊢ E We On | |
14 | wess 5656 | . . 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 7439 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ V) | |
17 | cantnfcl.g | . . . . . 6 ⊢ 𝐺 = OrdIso( E , (𝐹 supp ∅)) | |
18 | 17 | oion 9530 | . . . . 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 9368 | . . . . 5 ⊢ (𝜑 → (𝐹 supp ∅) ∈ Fin) |
22 | 17 | oien 9532 | . . . . . 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 9188 | . . . . 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 4189 | . . 3 ⊢ (𝜑 → dom 𝐺 ∈ (On ∩ Fin)) |
27 | onfin2 9230 | . . 3 ⊢ ω = (On ∩ Fin) | |
28 | 26, 27 | eleqtrrdi 2838 | . 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 1533 ∈ wcel 2098 Vcvv 3468 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 E cep 5572 We wwe 5623 dom cdm 5669 Oncon0 6357 ⟶wf 6532 (class class class)co 7404 ωcom 7851 supp csupp 8143 ≈ cen 8935 Fincfn 8938 finSupp cfsupp 9360 OrdIsocoi 9503 CNF ccnf 9655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-seqom 8446 df-1o 8464 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-oi 9504 df-cnf 9656 |
This theorem is referenced by: cantnfval2 9663 cantnfle 9665 cantnflt 9666 cantnflt2 9667 cantnff 9668 cantnfp1lem2 9673 cantnfp1lem3 9674 cantnflem1b 9680 cantnflem1d 9682 cantnflem1 9683 cnfcomlem 9693 cnfcom 9694 cnfcom2lem 9695 cnfcom3lem 9697 |
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