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Mirrors > Home > MPE Home > Th. List > ordtypelem8 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 8980. (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 8969 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
9 | 8 | fdmd 6497 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) |
10 | inss1 4155 | . . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇 | |
11 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem2 8967 | . . . . . 6 ⊢ (𝜑 → Ord 𝑇) |
12 | ordsson 7484 | . . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On) |
14 | 10, 13 | sstrid 3926 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On) |
15 | 9, 14 | eqsstrd 3953 | . . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On) |
16 | epweon 7477 | . . . 4 ⊢ E We On | |
17 | weso 5510 | . . . 4 ⊢ ( E We On → E Or On) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ E Or On |
19 | soss 5457 | . . 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 6494 | . . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴) |
22 | wess 5506 | . . . 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 5510 | . . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂) | |
25 | sopo 5456 | . . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂) | |
26 | 23, 24, 25 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂) |
27 | 8 | ffund 6491 | . . 3 ⊢ (𝜑 → Fun 𝑂) |
28 | funforn 6572 | . . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂) | |
29 | 27, 28 | sylib 221 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂) |
30 | epel 5433 | . . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏) | |
31 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem6 8971 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
32 | 30, 31 | syl5bi 245 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
33 | 32 | ralrimiva 3149 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
34 | 33 | ralrimivw 3150 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
35 | soisoi 7060 | . 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 837 | 1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 {crab 3110 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 ↦ cmpt 5110 E cep 5429 Po wpo 5436 Or wor 5437 Se wse 5476 We wwe 5477 dom cdm 5519 ran crn 5520 “ cima 5522 Ord word 6158 Oncon0 6159 Fun wfun 6318 –onto→wfo 6322 ‘cfv 6324 Isom wiso 6325 ℩crio 7092 recscrecs 7990 OrdIsocoi 8957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-wrecs 7930 df-recs 7991 df-oi 8958 |
This theorem is referenced by: ordtypelem9 8974 ordtypelem10 8975 oiiso2 8979 |
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