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Mirrors > Home > MPE Home > Th. List > cantnffval2 | Structured version Visualization version GIF version |
Description: An alternate definition of df-cnf 9700 which relies on cantnf 9731. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9702 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 9731 | . . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵))) |
6 | isof1o 7343 | . . . 4 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵)) | |
7 | f1orel 6852 | . . . 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 6211 | . . 3 ⊢ (Rel (𝐴 CNF 𝐵) ↔ ◡◡(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵)) | |
10 | 8, 9 | sylib 218 | . 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = (𝐴 CNF 𝐵)) |
11 | oecl 8574 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
12 | 2, 3, 11 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
13 | eloni 6396 | . . . . . 6 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑o 𝐵)) |
15 | isocnv 7350 | . . . . . 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 9732 | . . . . . . 7 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))) |
18 | 17 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝑇 We 𝑆) |
19 | ovex 7464 | . . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V | |
20 | 19 | dmex 7932 | . . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V |
21 | 1, 20 | eqeltri 2835 | . . . . . . 7 ⊢ 𝑆 ∈ V |
22 | exse 5649 | . . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆) | |
23 | 21, 22 | ax-mp 5 | . . . . . 6 ⊢ 𝑇 Se 𝑆 |
24 | eqid 2735 | . . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆) | |
25 | 24 | oieu 9577 | . . . . . 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 586 | . . . . 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 712 | . . . 4 ⊢ (𝜑 → ((𝐴 ↑o 𝐵) = dom OrdIso(𝑇, 𝑆) ∧ ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆))) |
28 | 27 | simprd 495 | . . 3 ⊢ (𝜑 → ◡(𝐴 CNF 𝐵) = OrdIso(𝑇, 𝑆)) |
29 | 28 | cnveqd 5889 | . 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
30 | 10, 29 | eqtr3d 2777 | 1 ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 Vcvv 3478 {copab 5210 E cep 5588 Se wse 5639 We wwe 5640 ◡ccnv 5688 dom cdm 5689 Rel wrel 5694 Ord word 6385 Oncon0 6386 –1-1-onto→wf1o 6562 ‘cfv 6563 Isom wiso 6564 (class class class)co 7431 ↑o coe 8504 OrdIsocoi 9547 CNF ccnf 9699 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-seqom 8487 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-oexp 8511 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-oi 9548 df-cnf 9700 |
This theorem is referenced by: (None) |
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