![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ordtypelem8 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 8793. (Contributed by Mario Carneiro, 25-Jun-2015.) |
Ref | Expression |
---|---|
ordtypelem.1 | ⊢ 𝐹 = recs(𝐺) |
ordtypelem.2 | ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} |
ordtypelem.3 | ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) |
ordtypelem.5 | ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} |
ordtypelem.6 | ⊢ 𝑂 = OrdIso(𝑅, 𝐴) |
ordtypelem.7 | ⊢ (𝜑 → 𝑅 We 𝐴) |
ordtypelem.8 | ⊢ (𝜑 → 𝑅 Se 𝐴) |
Ref | Expression |
---|---|
ordtypelem8 | ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtypelem.1 | . . . . . 6 ⊢ 𝐹 = recs(𝐺) | |
2 | ordtypelem.2 | . . . . . 6 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} | |
3 | ordtypelem.3 | . . . . . 6 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) | |
4 | ordtypelem.5 | . . . . . 6 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} | |
5 | ordtypelem.6 | . . . . . 6 ⊢ 𝑂 = OrdIso(𝑅, 𝐴) | |
6 | ordtypelem.7 | . . . . . 6 ⊢ (𝜑 → 𝑅 We 𝐴) | |
7 | ordtypelem.8 | . . . . . 6 ⊢ (𝜑 → 𝑅 Se 𝐴) | |
8 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem4 8782 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
9 | 8 | fdmd 6355 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) |
10 | inss1 4094 | . . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇 | |
11 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem2 8780 | . . . . . 6 ⊢ (𝜑 → Ord 𝑇) |
12 | ordsson 7322 | . . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On) |
14 | 10, 13 | syl5ss 3871 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On) |
15 | 9, 14 | eqsstrd 3897 | . . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On) |
16 | epweon 7315 | . . . 4 ⊢ E We On | |
17 | weso 5399 | . . . 4 ⊢ ( E We On → E Or On) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ E Or On |
19 | soss 5346 | . . 3 ⊢ (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂)) | |
20 | 15, 18, 19 | mpisyl 21 | . 2 ⊢ (𝜑 → E Or dom 𝑂) |
21 | 8 | frnd 6353 | . . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴) |
22 | wess 5395 | . . . 4 ⊢ (ran 𝑂 ⊆ 𝐴 → (𝑅 We 𝐴 → 𝑅 We ran 𝑂)) | |
23 | 21, 6, 22 | sylc 65 | . . 3 ⊢ (𝜑 → 𝑅 We ran 𝑂) |
24 | weso 5399 | . . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂) | |
25 | sopo 5345 | . . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂) | |
26 | 23, 24, 25 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂) |
27 | 8 | ffund 6350 | . . 3 ⊢ (𝜑 → Fun 𝑂) |
28 | funforn 6428 | . . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂) | |
29 | 27, 28 | sylib 210 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂) |
30 | epel 5322 | . . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏) | |
31 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem6 8784 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
32 | 30, 31 | syl5bi 234 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
33 | 32 | ralrimiva 3132 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
34 | 33 | ralrimivw 3133 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
35 | soisoi 6906 | . 2 ⊢ ((( E Or dom 𝑂 ∧ 𝑅 Po ran 𝑂) ∧ (𝑂:dom 𝑂–onto→ran 𝑂 ∧ ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏)))) → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) | |
36 | 20, 26, 29, 34, 35 | syl22anc 826 | 1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ∀wral 3088 ∃wrex 3089 {crab 3092 Vcvv 3415 ∩ cin 3830 ⊆ wss 3831 class class class wbr 4930 ↦ cmpt 5009 E cep 5317 Po wpo 5325 Or wor 5326 Se wse 5365 We wwe 5366 dom cdm 5408 ran crn 5409 “ cima 5411 Ord word 6030 Oncon0 6031 Fun wfun 6184 –onto→wfo 6188 ‘cfv 6190 Isom wiso 6191 ℩crio 6938 recscrecs 7813 OrdIsocoi 8770 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-se 5368 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-isom 6199 df-riota 6939 df-wrecs 7752 df-recs 7814 df-oi 8771 |
This theorem is referenced by: ordtypelem9 8787 ordtypelem10 8788 oiiso2 8792 |
Copyright terms: Public domain | W3C validator |