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Mirrors > Home > MPE Home > Th. List > ordtypelem5 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 9566. (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 9553 | . . . 4 ⊢ (𝜑 → Ord 𝑇) |
9 | 1 | tfr1a 8414 | . . . . . 6 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
10 | 9 | simpri 484 | . . . . 5 ⊢ Lim dom 𝐹 |
11 | limord 6426 | . . . . 5 ⊢ (Lim dom 𝐹 → Ord dom 𝐹) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ Ord dom 𝐹 |
13 | ordin 6396 | . . . 4 ⊢ ((Ord 𝑇 ∧ Ord dom 𝐹) → Ord (𝑇 ∩ dom 𝐹)) | |
14 | 8, 12, 13 | sylancl 584 | . . 3 ⊢ (𝜑 → Ord (𝑇 ∩ dom 𝐹)) |
15 | 1, 2, 3, 4, 5, 6, 7 | ordtypelem4 9555 | . . . . 5 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
16 | 15 | fdmd 6728 | . . . 4 ⊢ (𝜑 → dom 𝑂 = (𝑇 ∩ dom 𝐹)) |
17 | ordeq 6373 | . . . 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 256 | . 2 ⊢ (𝜑 → Ord dom 𝑂) |
20 | 15 | ffdmd 6749 | . 2 ⊢ (𝜑 → 𝑂:dom 𝑂⟶𝐴) |
21 | 19, 20 | jca 510 | 1 ⊢ (𝜑 → (Ord dom 𝑂 ∧ 𝑂:dom 𝑂⟶𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∀wral 3051 ∃wrex 3060 {crab 3420 Vcvv 3463 ∩ cin 3946 class class class wbr 5144 ↦ cmpt 5227 Se wse 5626 We wwe 5627 dom cdm 5673 ran crn 5674 “ cima 5676 Ord word 6365 Oncon0 6366 Lim wlim 6367 Fun wfun 6538 ⟶wf 6540 ℩crio 7369 recscrecs 8390 OrdIsocoi 9543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pr 5424 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-oi 9544 |
This theorem is referenced by: oicl 9563 oif 9564 |
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