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Mirrors > Home > MPE Home > Th. List > ordtypelem5 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 8437. (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 8424 | . . . 4 ⊢ (𝜑 → Ord 𝑇) |
9 | 1 | tfr1a 7490 | . . . . . 6 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
10 | 9 | simpri 478 | . . . . 5 ⊢ Lim dom 𝐹 |
11 | limord 5784 | . . . . 5 ⊢ (Lim dom 𝐹 → Ord dom 𝐹) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ Ord dom 𝐹 |
13 | ordin 5753 | . . . 4 ⊢ ((Ord 𝑇 ∧ Ord dom 𝐹) → Ord (𝑇 ∩ dom 𝐹)) | |
14 | 8, 12, 13 | sylancl 694 | . . 3 ⊢ (𝜑 → Ord (𝑇 ∩ dom 𝐹)) |
15 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem4 8426 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
16 | fdm 6051 | . . . . 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 5730 | . . . 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 247 | . 2 ⊢ (𝜑 → Ord dom 𝑂) |
21 | 17 | feq2d 6031 | . . 3 ⊢ (𝜑 → (𝑂:dom 𝑂⟶𝐴 ↔ 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴)) |
22 | 15, 21 | mpbird 247 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂⟶𝐴) |
23 | 20, 22 | jca 554 | 1 ⊢ (𝜑 → (Ord dom 𝑂 ∧ 𝑂:dom 𝑂⟶𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∩ cin 3573 class class class wbr 4653 ↦ cmpt 4729 Se wse 5071 We wwe 5072 dom cdm 5114 ran crn 5115 “ cima 5117 Ord word 5722 Oncon0 5723 Lim wlim 5724 Fun wfun 5882 ⟶wf 5884 ℩crio 6610 recscrecs 7467 OrdIsocoi 8414 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-wrecs 7407 df-recs 7468 df-oi 8415 |
This theorem is referenced by: oicl 8434 oif 8435 |
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