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| Mirrors > Home > MPE Home > Th. List > ordtypelem1 | Structured version Visualization version GIF version | ||
| Description: Lemma for ordtype 9572. (Contributed by Mario Carneiro, 24-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 |
|---|---|
| ordtypelem1 | ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ordtypelem.7 | . . 3 ⊢ (𝜑 → 𝑅 We 𝐴) | |
| 2 | ordtypelem.8 | . . 3 ⊢ (𝜑 → 𝑅 Se 𝐴) | |
| 3 | iftrue 4531 | . . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}), ∅) = (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡})) | |
| 4 | 1, 2, 3 | syl2anc 584 | . 2 ⊢ (𝜑 → if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}), ∅) = (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡})) |
| 5 | ordtypelem.6 | . . 3 ⊢ 𝑂 = OrdIso(𝑅, 𝐴) | |
| 6 | ordtypelem.2 | . . . 4 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} | |
| 7 | ordtypelem.3 | . . . 4 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) | |
| 8 | ordtypelem.1 | . . . 4 ⊢ 𝐹 = recs(𝐺) | |
| 9 | 6, 7, 8 | dfoi 9551 | . . 3 ⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}), ∅) |
| 10 | 5, 9 | eqtri 2765 | . 2 ⊢ 𝑂 = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}), ∅) |
| 11 | ordtypelem.5 | . . 3 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} | |
| 12 | 11 | reseq2i 5994 | . 2 ⊢ (𝐹 ↾ 𝑇) = (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}) |
| 13 | 4, 10, 12 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∀wral 3061 ∃wrex 3070 {crab 3436 Vcvv 3480 ∅c0 4333 ifcif 4525 class class class wbr 5143 ↦ cmpt 5225 Se wse 5635 We wwe 5636 ran crn 5686 ↾ cres 5687 “ cima 5688 Oncon0 6384 ℩crio 7387 recscrecs 8410 OrdIsocoi 9549 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-xp 5691 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fv 6569 df-riota 7388 df-ov 7434 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-oi 9550 |
| This theorem is referenced by: ordtypelem4 9561 ordtypelem6 9563 ordtypelem7 9564 ordtypelem9 9566 |
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