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Mirrors > Home > MPE Home > Th. List > ordtypelem8 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 8990. (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 8979 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
9 | 8 | fdmd 6517 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) |
10 | inss1 4204 | . . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇 | |
11 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem2 8977 | . . . . . 6 ⊢ (𝜑 → Ord 𝑇) |
12 | ordsson 7498 | . . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On) |
14 | 10, 13 | sstrid 3977 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On) |
15 | 9, 14 | eqsstrd 4004 | . . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On) |
16 | epweon 7491 | . . . 4 ⊢ E We On | |
17 | weso 5540 | . . . 4 ⊢ ( E We On → E Or On) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ E Or On |
19 | soss 5487 | . . 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 6515 | . . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴) |
22 | wess 5536 | . . . 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 5540 | . . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂) | |
25 | sopo 5486 | . . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂) | |
26 | 23, 24, 25 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂) |
27 | 8 | ffund 6512 | . . 3 ⊢ (𝜑 → Fun 𝑂) |
28 | funforn 6591 | . . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂) | |
29 | 27, 28 | sylib 220 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂) |
30 | epel 5463 | . . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏) | |
31 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem6 8981 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
32 | 30, 31 | syl5bi 244 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
33 | 32 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
34 | 33 | ralrimivw 3183 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
35 | soisoi 7075 | . 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 836 | 1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∩ cin 3934 ⊆ wss 3935 class class class wbr 5058 ↦ cmpt 5138 E cep 5458 Po wpo 5466 Or wor 5467 Se wse 5506 We wwe 5507 dom cdm 5549 ran crn 5550 “ cima 5552 Ord word 6184 Oncon0 6185 Fun wfun 6343 –onto→wfo 6347 ‘cfv 6349 Isom wiso 6350 ℩crio 7107 recscrecs 8001 OrdIsocoi 8967 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-wrecs 7941 df-recs 8002 df-oi 8968 |
This theorem is referenced by: ordtypelem9 8984 ordtypelem10 8985 oiiso2 8989 |
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