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Mirrors > Home > MPE Home > Th. List > ordtypelem1 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 8842. (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 4387 | . . 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 8821 | . . 3 ⊢ OrdIso(𝑅, 𝐴) = if((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴), (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}), ∅) |
10 | 5, 9 | eqtri 2819 | . 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 5731 | . 2 ⊢ (𝐹 ↾ 𝑇) = (𝐹 ↾ {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡}) |
13 | 4, 10, 12 | 3eqtr4g 2856 | 1 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∀wral 3105 ∃wrex 3106 {crab 3109 Vcvv 3437 ∅c0 4211 ifcif 4381 class class class wbr 4962 ↦ cmpt 5041 Se wse 5400 We wwe 5401 ran crn 5444 ↾ cres 5445 “ cima 5446 Oncon0 6066 ℩crio 6976 recscrecs 7859 OrdIsocoi 8819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-xp 5449 df-cnv 5451 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-iota 6189 df-fv 6233 df-riota 6977 df-wrecs 7798 df-recs 7860 df-oi 8820 |
This theorem is referenced by: ordtypelem4 8831 ordtypelem6 8833 ordtypelem7 8834 ordtypelem9 8836 |
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