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Mirrors > Home > MPE Home > Th. List > ordtypelem5 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 8320. (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 8307 | . . . 4 ⊢ (𝜑 → Ord 𝑇) |
9 | 1 | tfr1a 7377 | . . . . . 6 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
10 | 9 | simpri 477 | . . . . 5 ⊢ Lim dom 𝐹 |
11 | limord 5701 | . . . . 5 ⊢ (Lim dom 𝐹 → Ord dom 𝐹) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ Ord dom 𝐹 |
13 | ordin 5670 | . . . 4 ⊢ ((Ord 𝑇 ∧ Ord dom 𝐹) → Ord (𝑇 ∩ dom 𝐹)) | |
14 | 8, 12, 13 | sylancl 693 | . . 3 ⊢ (𝜑 → Ord (𝑇 ∩ dom 𝐹)) |
15 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem4 8309 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
16 | fdm 5964 | . . . . 5 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) | |
17 | 15, 16 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) |
18 | ordeq 5647 | . . . 4 ⊢ (dom 𝑂 = (𝑇 ∩ dom 𝐹) → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹))) | |
19 | 17, 18 | syl 17 | . . 3 ⊢ (𝜑 → (Ord dom 𝑂 ↔ Ord (𝑇 ∩ dom 𝐹))) |
20 | 14, 19 | mpbird 246 | . 2 ⊢ (𝜑 → Ord dom 𝑂) |
21 | 17 | feq2d 5944 | . . 3 ⊢ (𝜑 → (𝑂:dom 𝑂⟶𝐴 ↔ 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)) |
22 | 15, 21 | mpbird 246 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂⟶𝐴) |
23 | 20, 22 | jca 553 | 1 ⊢ (𝜑 → (Ord dom 𝑂 ∧ 𝑂:dom 𝑂⟶𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∩ cin 3539 class class class wbr 4583 ↦ cmpt 4643 Se wse 4995 We wwe 4996 dom cdm 5038 ran crn 5039 “ cima 5041 Ord word 5639 Oncon0 5640 Lim wlim 5641 Fun wfun 5798 ⟶wf 5800 ℩crio 6510 recscrecs 7354 OrdIsocoi 8297 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-wrecs 7294 df-recs 7355 df-oi 8298 |
This theorem is referenced by: oicl 8317 oif 8318 |
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