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Mirrors > Home > MPE Home > Th. List > cantnffval2 | Structured version Visualization version GIF version |
Description: An alternate definition of df-cnf 9693 which relies on cantnf 9724. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9695 to eliminate it.) (Contributed by Mario Carneiro, 28-May-2015.) |
Ref | Expression |
---|---|
cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
oemapval.t | ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} |
Ref | Expression |
---|---|
cantnffval2 | ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cantnfs.s | . . . . 5 ⊢ 𝑆 = dom (𝐴 CNF 𝐵) | |
2 | cantnfs.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ On) | |
3 | cantnfs.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ On) | |
4 | oemapval.t | . . . . 5 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} | |
5 | 1, 2, 3, 4 | cantnf 9724 | . . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵))) |
6 | isof1o 7337 | . . . 4 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵)) | |
7 | f1orel 6847 | . . . 4 ⊢ ((𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵) → Rel (𝐴 CNF 𝐵)) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → Rel (𝐴 CNF 𝐵)) |
9 | dfrel2 6198 | . . 3 ⊢ (Rel (𝐴 CNF 𝐵) ↔ ◡◡(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵)) | |
10 | 8, 9 | sylib 217 | . 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵)) |
11 | oecl 8564 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
12 | 2, 3, 11 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
13 | eloni 6384 | . . . . . 6 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑o 𝐵)) |
15 | isocnv 7344 | . . . . . 6 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) | |
16 | 5, 15 | syl 17 | . . . . 5 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) |
17 | 1, 2, 3, 4 | oemapwe 9725 | . . . . . . 7 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))) |
18 | 17 | simpld 493 | . . . . . 6 ⊢ (𝜑 → 𝑇 We 𝑆) |
19 | ovex 7459 | . . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V | |
20 | 19 | dmex 7923 | . . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V |
21 | 1, 20 | eqeltri 2825 | . . . . . . 7 ⊢ 𝑆 ∈ V |
22 | exse 5645 | . . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆) | |
23 | 21, 22 | ax-mp 5 | . . . . . 6 ⊢ 𝑇 Se 𝑆 |
24 | eqid 2728 | . . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆) | |
25 | 24 | oieu 9570 | . . . . . 6 ⊢ ((𝑇 We 𝑆 ∧ 𝑇 Se 𝑆) → ((Ord (𝐴 ↑o 𝐵) ∧ ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) ↔ ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))) |
26 | 18, 23, 25 | sylancl 584 | . . . . 5 ⊢ (𝜑 → ((Ord (𝐴 ↑o 𝐵) ∧ ◡(𝐴 CNF 𝐵) Isom E , 𝑇 ((𝐴 ↑o 𝐵), 𝑆)) ↔ ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)))) |
27 | 14, 16, 26 | mpbi2and 710 | . . . 4 ⊢ (𝜑 → ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))) |
28 | 27 | simprd 494 | . . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)) |
29 | 28 | cnveqd 5882 | . 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
30 | 10, 29 | eqtr3d 2770 | 1 ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 Vcvv 3473 {copab 5214 E cep 5585 Se wse 5635 We wwe 5636 ◡ccnv 5681 dom cdm 5682 Rel wrel 5687 Ord word 6373 Oncon0 6374 –1-1-onto→wf1o 6552 ‘cfv 6553 Isom wiso 6554 (class class class)co 7426 ↑o coe 8492 OrdIsocoi 9540 CNF ccnf 9692 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-supp 8172 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-seqom 8475 df-1o 8493 df-2o 8494 df-oadd 8497 df-omul 8498 df-oexp 8499 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-fsupp 9394 df-oi 9541 df-cnf 9693 |
This theorem is referenced by: (None) |
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