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| Mirrors > Home > MPE Home > Th. List > ordtypelem8 | Structured version Visualization version GIF version | ||
| Description: Lemma for ordtype 9572. (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 9561 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
| 9 | 8 | fdmd 6746 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) |
| 10 | inss1 4237 | . . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇 | |
| 11 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem2 9559 | . . . . . 6 ⊢ (𝜑 → Ord 𝑇) |
| 12 | ordsson 7803 | . . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On) | |
| 13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On) |
| 14 | 10, 13 | sstrid 3995 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On) |
| 15 | 9, 14 | eqsstrd 4018 | . . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On) |
| 16 | epweon 7795 | . . . 4 ⊢ E We On | |
| 17 | weso 5676 | . . . 4 ⊢ ( E We On → E Or On) | |
| 18 | 16, 17 | ax-mp 5 | . . 3 ⊢ E Or On |
| 19 | soss 5612 | . . 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 6744 | . . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴) |
| 22 | wess 5671 | . . . 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 5676 | . . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂) | |
| 25 | sopo 5611 | . . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂) | |
| 26 | 23, 24, 25 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂) |
| 27 | 8 | ffund 6740 | . . 3 ⊢ (𝜑 → Fun 𝑂) |
| 28 | funforn 6827 | . . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂) | |
| 29 | 27, 28 | sylib 218 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂) |
| 30 | epel 5587 | . . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏) | |
| 31 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem6 9563 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
| 32 | 30, 31 | biimtrid 242 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
| 33 | 32 | ralrimiva 3146 | . . 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 7348 | . 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 839 | 1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 class class class wbr 5143 ↦ cmpt 5225 E cep 5583 Po wpo 5590 Or wor 5591 Se wse 5635 We wwe 5636 dom cdm 5685 ran crn 5686 “ cima 5688 Ord word 6383 Oncon0 6384 Fun wfun 6555 –onto→wfo 6559 ‘cfv 6561 Isom wiso 6562 ℩crio 7387 recscrecs 8410 OrdIsocoi 9549 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-oi 9550 |
| This theorem is referenced by: ordtypelem9 9566 ordtypelem10 9567 oiiso2 9571 |
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