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| Mirrors > Home > MPE Home > Th. List > ordtypelem5 | 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 | 
|---|---|
| ordtypelem5 | ⊢ (𝜑 → (Ord dom 𝑂 ∧ 𝑂:dom 𝑂⟶𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ordtypelem.1 | . . . . 5 ⊢ 𝐹 = recs(𝐺) | |
| 2 | ordtypelem.2 | . . . . 5 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} | |
| 3 | ordtypelem.3 | . . . . 5 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) | |
| 4 | ordtypelem.5 | . . . . 5 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} | |
| 5 | ordtypelem.6 | . . . . 5 ⊢ 𝑂 = OrdIso(𝑅, 𝐴) | |
| 6 | ordtypelem.7 | . . . . 5 ⊢ (𝜑 → 𝑅 We 𝐴) | |
| 7 | ordtypelem.8 | . . . . 5 ⊢ (𝜑 → 𝑅 Se 𝐴) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem2 9559 | . . . 4 ⊢ (𝜑 → Ord 𝑇) | 
| 9 | 1 | tfr1a 8434 | . . . . . 6 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) | 
| 10 | 9 | simpri 485 | . . . . 5 ⊢ Lim dom 𝐹 | 
| 11 | limord 6444 | . . . . 5 ⊢ (Lim dom 𝐹 → Ord dom 𝐹) | |
| 12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ Ord dom 𝐹 | 
| 13 | ordin 6414 | . . . 4 ⊢ ((Ord 𝑇 ∧ Ord dom 𝐹) → Ord (𝑇 ∩ dom 𝐹)) | |
| 14 | 8, 12, 13 | sylancl 586 | . . 3 ⊢ (𝜑 → Ord (𝑇 ∩ dom 𝐹)) | 
| 15 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem4 9561 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) | 
| 16 | 15 | fdmd 6746 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) | 
| 17 | ordeq 6391 | . . . 4 ⊢ (dom 𝑂 = (𝑇 ∩ dom 𝐹) → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹))) | |
| 18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹))) | 
| 19 | 14, 18 | mpbird 257 | . 2 ⊢ (𝜑 → Ord dom 𝑂) | 
| 20 | 15 | ffdmd 6766 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂⟶𝐴) | 
| 21 | 19, 20 | jca 511 | 1 ⊢ (𝜑 → (Ord dom 𝑂 ∧ 𝑂:dom 𝑂⟶𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∩ cin 3950 class class class wbr 5143 ↦ cmpt 5225 Se wse 5635 We wwe 5636 dom cdm 5685 ran crn 5686 “ cima 5688 Ord word 6383 Oncon0 6384 Lim wlim 6385 Fun wfun 6555 ⟶wf 6557 ℩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-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: oicl 9569 oif 9570 | 
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