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Mirrors > Home > MPE Home > Th. List > ordtypelem8 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 9557. (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 9546 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
9 | 8 | fdmd 6733 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) |
10 | inss1 4227 | . . . . 5 ⊢ (𝑇 ∩ dom 𝐹) ⊆ 𝑇 | |
11 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem2 9544 | . . . . . 6 ⊢ (𝜑 → Ord 𝑇) |
12 | ordsson 7786 | . . . . . 6 ⊢ (Ord 𝑇 → 𝑇 ⊆ On) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑇 ⊆ On) |
14 | 10, 13 | sstrid 3988 | . . . 4 ⊢ (𝜑 → (𝑇 ∩ dom 𝐹) ⊆ On) |
15 | 9, 14 | eqsstrd 4015 | . . 3 ⊢ (𝜑 → dom 𝑂 ⊆ On) |
16 | epweon 7778 | . . . 4 ⊢ E We On | |
17 | weso 5669 | . . . 4 ⊢ ( E We On → E Or On) | |
18 | 16, 17 | ax-mp 5 | . . 3 ⊢ E Or On |
19 | soss 5610 | . . 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 6731 | . . . 4 ⊢ (𝜑 → ran 𝑂 ⊆ 𝐴) |
22 | wess 5665 | . . . 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 5669 | . . 3 ⊢ (𝑅 We ran 𝑂 → 𝑅 Or ran 𝑂) | |
25 | sopo 5609 | . . 3 ⊢ (𝑅 Or ran 𝑂 → 𝑅 Po ran 𝑂) | |
26 | 23, 24, 25 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑅 Po ran 𝑂) |
27 | 8 | ffund 6727 | . . 3 ⊢ (𝜑 → Fun 𝑂) |
28 | funforn 6817 | . . 3 ⊢ (Fun 𝑂 ↔ 𝑂:dom 𝑂–onto→ran 𝑂) | |
29 | 27, 28 | sylib 217 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂–onto→ran 𝑂) |
30 | epel 5585 | . . . . 5 ⊢ (𝑎 E 𝑏 ↔ 𝑎 ∈ 𝑏) | |
31 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem6 9548 | . . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 ∈ 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
32 | 30, 31 | biimtrid 241 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ dom 𝑂) → (𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
33 | 32 | ralrimiva 3135 | . . 3 ⊢ (𝜑 → ∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
34 | 33 | ralrimivw 3139 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑂∀𝑏 ∈ dom 𝑂(𝑎 E 𝑏 → (𝑂‘𝑎)𝑅(𝑂‘𝑏))) |
35 | soisoi 7335 | . 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 837 | 1 ⊢ (𝜑 → 𝑂 Isom E , 𝑅 (dom 𝑂, ran 𝑂)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 {crab 3418 Vcvv 3461 ∩ cin 3943 ⊆ wss 3944 class class class wbr 5149 ↦ cmpt 5232 E cep 5581 Po wpo 5588 Or wor 5589 Se wse 5631 We wwe 5632 dom cdm 5678 ran crn 5679 “ cima 5681 Ord word 6370 Oncon0 6371 Fun wfun 6543 –onto→wfo 6547 ‘cfv 6549 Isom wiso 6550 ℩crio 7374 recscrecs 8391 OrdIsocoi 9534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-oi 9535 |
This theorem is referenced by: ordtypelem9 9551 ordtypelem10 9552 oiiso2 9556 |
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