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Mirrors > Home > MPE Home > Th. List > cantnffval2 | Structured version Visualization version GIF version |
Description: An alternate definition of df-cnf 8442 which relies on cantnf 8473. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 8444 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 8473 | . . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑𝑜 𝐵))) |
6 | isof1o 6473 | . . . 4 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑𝑜 𝐵)) → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑𝑜 𝐵)) | |
7 | f1orel 6053 | . . . 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 5502 | . . 3 ⊢ (Rel (𝐴 CNF 𝐵) ↔ ◡◡(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵)) | |
10 | 8, 9 | sylib 207 | . 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵)) |
11 | oecl 7504 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On) | |
12 | 2, 3, 11 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐵) ∈ On) |
13 | eloni 5650 | . . . . . 6 ⊢ ((𝐴 ↑𝑜 𝐵) ∈ On → Ord (𝐴 ↑𝑜 𝐵)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑𝑜 𝐵)) |
15 | isocnv 6480 | . . . . . 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 8474 | . . . . . . 7 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑𝑜 𝐵))) |
18 | 17 | simpld 474 | . . . . . 6 ⊢ (𝜑 → 𝑇 We 𝑆) |
19 | ovex 6577 | . . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V | |
20 | 19 | dmex 6991 | . . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V |
21 | 1, 20 | eqeltri 2684 | . . . . . . 7 ⊢ 𝑆 ∈ V |
22 | exse 5002 | . . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆) | |
23 | 21, 22 | ax-mp 5 | . . . . . 6 ⊢ 𝑇 Se 𝑆 |
24 | eqid 2610 | . . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆) | |
25 | 24 | oieu 8327 | . . . . . 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 693 | . . . . 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 958 | . . . 4 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))) |
28 | 27 | simprd 478 | . . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)) |
29 | 28 | cnveqd 5220 | . 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
30 | 10, 29 | eqtr3d 2646 | 1 ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 Vcvv 3173 {copab 4642 E cep 4947 Se wse 4995 We wwe 4996 ◡ccnv 5037 dom cdm 5038 Rel wrel 5043 Ord word 5639 Oncon0 5640 –1-1-onto→wf1o 5803 ‘cfv 5804 Isom wiso 5805 (class class class)co 6549 ↑𝑜 coe 7446 OrdIsocoi 8297 CNF ccnf 8441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-oexp 7453 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-oi 8298 df-cnf 8442 |
This theorem is referenced by: (None) |
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