![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > cantnffval2 | Structured version Visualization version GIF version |
Description: An alternate definition of df-cnf 9659 which relies on cantnf 9690. (Note that although the use of 𝑆 seems self-referential, one can use cantnfdm 9661 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 9690 | . . . 4 ⊢ (𝜑 → (𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵))) |
6 | isof1o 7316 | . . . 4 ⊢ ((𝐴 CNF 𝐵) Isom 𝑇, E (𝑆, (𝐴 ↑o 𝐵)) → (𝐴 CNF 𝐵):𝑆–1-1-onto→(𝐴 ↑o 𝐵)) | |
7 | f1orel 6830 | . . . 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 6182 | . . 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 8538 | . . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
12 | 2, 3, 11 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
13 | eloni 6368 | . . . . . 6 ⊢ ((𝐴 ↑o 𝐵) ∈ On → Ord (𝐴 ↑o 𝐵)) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → Ord (𝐴 ↑o 𝐵)) |
15 | isocnv 7323 | . . . . . 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 9691 | . . . . . . 7 ⊢ (𝜑 → (𝑇 We 𝑆 ∧ dom OrdIso(𝑇, 𝑆) = (𝐴 ↑o 𝐵))) |
18 | 17 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝑇 We 𝑆) |
19 | ovex 7438 | . . . . . . . . 9 ⊢ (𝐴 CNF 𝐵) ∈ V | |
20 | 19 | dmex 7899 | . . . . . . . 8 ⊢ dom (𝐴 CNF 𝐵) ∈ V |
21 | 1, 20 | eqeltri 2823 | . . . . . . 7 ⊢ 𝑆 ∈ V |
22 | exse 5632 | . . . . . . 7 ⊢ (𝑆 ∈ V → 𝑇 Se 𝑆) | |
23 | 21, 22 | ax-mp 5 | . . . . . 6 ⊢ 𝑇 Se 𝑆 |
24 | eqid 2726 | . . . . . . 7 ⊢ OrdIso(𝑇, 𝑆) = OrdIso(𝑇, 𝑆) | |
25 | 24 | oieu 9536 | . . . . . 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 585 | . . . . 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 709 | . . . 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 5869 | . 2 ⊢ (𝜑 → ◡◡(𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
30 | 10, 29 | eqtr3d 2768 | 1 ⊢ (𝜑 → (𝐴 CNF 𝐵) = ◡OrdIso(𝑇, 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 Vcvv 3468 {copab 5203 E cep 5572 Se wse 5622 We wwe 5623 ◡ccnv 5668 dom cdm 5669 Rel wrel 5674 Ord word 6357 Oncon0 6358 –1-1-onto→wf1o 6536 ‘cfv 6537 Isom wiso 6538 (class class class)co 7405 ↑o coe 8466 OrdIsocoi 9506 CNF ccnf 9658 |
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 7722 |
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-int 4944 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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-seqom 8449 df-1o 8467 df-2o 8468 df-oadd 8471 df-omul 8472 df-oexp 8473 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-oi 9507 df-cnf 9659 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |