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Mirrors > Home > MPE Home > Th. List > ordtypelem8 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 8593. (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 8582 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
9 | fdm 6191 | . . . . 5 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) |
11 | inss1 3981 | . . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇 | |
12 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem2 8580 | . . . . . 6 ⊢ (𝜑 → Ord 𝑇) |
13 | ordsson 7136 | . . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On) |
15 | 11, 14 | syl5ss 3763 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On) |
16 | 10, 15 | eqsstrd 3788 | . . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On) |
17 | epweon 7130 | . . . 4 ⊢ E We On | |
18 | weso 5240 | . . . 4 ⊢ ( E We On → E Or On) | |
19 | 17, 18 | ax-mp 5 | . . 3 ⊢ E Or On |
20 | soss 5188 | . . 3 ⊢ (dom 𝑂 ⊆ On → ( E Or On → E Or dom 𝑂)) | |
21 | 16, 19, 20 | mpisyl 21 | . 2 ⊢ (𝜑 → E Or dom 𝑂) |
22 | frn 6193 | . . . . 5 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → ran 𝑂 ⊆ 𝐴) | |
23 | 8, 22 | syl 17 | . . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴) |
24 | wess 5236 | . . . 4 ⊢ (ran 𝑂 ⊆ 𝐴 → (𝑅 We 𝐴 → 𝑅 We ran 𝑂)) | |
25 | 23, 6, 24 | sylc 65 | . . 3 ⊢ (𝜑 → 𝑅 We ran 𝑂) |
26 | weso 5240 | . . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂) | |
27 | sopo 5187 | . . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂) | |
28 | 25, 26, 27 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂) |
29 | ffun 6188 | . . . 4 ⊢ (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 → Fun 𝑂) | |
30 | 8, 29 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝑂) |
31 | funforn 6263 | . . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂) | |
32 | 30, 31 | sylib 208 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂) |
33 | epel 5165 | . . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏) | |
34 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem6 8584 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
35 | 33, 34 | syl5bi 232 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
36 | 35 | ralrimiva 3115 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
37 | 36 | ralrimivw 3116 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
38 | soisoi 6721 | . 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)) | |
39 | 21, 28, 32, 37, 38 | syl22anc 1477 | 1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∀wral 3061 ∃wrex 3062 {crab 3065 Vcvv 3351 ∩ cin 3722 ⊆ wss 3723 class class class wbr 4786 ↦ cmpt 4863 E cep 5161 Po wpo 5168 Or wor 5169 Se wse 5206 We wwe 5207 dom cdm 5249 ran crn 5250 “ cima 5252 Ord word 5865 Oncon0 5866 Fun wfun 6025 ⟶wf 6027 –onto→wfo 6029 ‘cfv 6031 Isom wiso 6032 ℩crio 6753 recscrecs 7620 OrdIsocoi 8570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-wrecs 7559 df-recs 7621 df-oi 8571 |
This theorem is referenced by: ordtypelem9 8587 ordtypelem10 8588 oiiso2 8592 |
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