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Mirrors > Home > MPE Home > Th. List > ordtypelem4 | 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 |
---|---|
ordtypelem4 | ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtypelem.1 | . . . . . . . 8 ⊢ 𝐹 = recs(𝐺) | |
2 | 1 | tfr1a 7882 | . . . . . . 7 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
3 | 2 | simpli 484 | . . . . . 6 ⊢ Fun 𝐹 |
4 | funres 6267 | . . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑇)) | |
5 | 3, 4 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑇)) |
6 | 5 | funfnd 6256 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇)) |
7 | dmres 5756 | . . . . 5 ⊢ dom (𝐹 ↾ 𝑇) = (𝑇 ∩ dom 𝐹) | |
8 | 7 | fneq2i 6321 | . . . 4 ⊢ ((𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇) ↔ (𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹)) |
9 | 6, 8 | sylib 219 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹)) |
10 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹)) | |
11 | 10 | elin1d 4096 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ 𝑇) |
12 | 11 | fvresd 6558 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) = (𝐹‘𝑎)) |
13 | ssrab2 3977 | . . . . . . 7 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} | |
14 | ssrab2 3977 | . . . . . . 7 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ⊆ 𝐴 | |
15 | 13, 14 | sstri 3898 | . . . . . 6 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ 𝐴 |
16 | ordtypelem.2 | . . . . . . 7 ⊢ 𝐶 = {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ ran ℎ 𝑗𝑅𝑤} | |
17 | ordtypelem.3 | . . . . . . 7 ⊢ 𝐺 = (ℎ ∈ V ↦ (℩𝑣 ∈ 𝐶 ∀𝑢 ∈ 𝐶 ¬ 𝑢𝑅𝑣)) | |
18 | ordtypelem.5 | . . . . . . 7 ⊢ 𝑇 = {𝑥 ∈ On ∣ ∃𝑡 ∈ 𝐴 ∀𝑧 ∈ (𝐹 “ 𝑥)𝑧𝑅𝑡} | |
19 | ordtypelem.6 | . . . . . . 7 ⊢ 𝑂 = OrdIso(𝑅, 𝐴) | |
20 | ordtypelem.7 | . . . . . . 7 ⊢ (𝜑 → 𝑅 We 𝐴) | |
21 | ordtypelem.8 | . . . . . . 7 ⊢ (𝜑 → 𝑅 Se 𝐴) | |
22 | 1, 16, 17, 18, 19, 20, 21 | ordtypelem3 8830 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑎) ∈ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣}) |
23 | 15, 22 | sseldi 3887 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑎) ∈ 𝐴) |
24 | 12, 23 | eqeltrd 2883 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴) |
25 | 24 | ralrimiva 3149 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴) |
26 | ffnfv 6745 | . . 3 ⊢ ((𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ ((𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹) ∧ ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴)) | |
27 | 9, 25, 26 | sylanbrc 583 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴) |
28 | 1, 16, 17, 18, 19, 20, 21 | ordtypelem1 8828 | . . 3 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) |
29 | 28 | feq1d 6367 | . 2 ⊢ (𝜑 → (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴)) |
30 | 27, 29 | mpbird 258 | 1 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∀wral 3105 ∃wrex 3106 {crab 3109 Vcvv 3437 ∩ cin 3858 class class class wbr 4962 ↦ cmpt 5041 Se wse 5400 We wwe 5401 dom cdm 5443 ran crn 5444 ↾ cres 5445 “ cima 5446 Oncon0 6066 Lim wlim 6067 Fun wfun 6219 Fn wfn 6220 ⟶wf 6221 ‘cfv 6225 ℩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-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-se 5403 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-wrecs 7798 df-recs 7860 df-oi 8820 |
This theorem is referenced by: ordtypelem5 8832 ordtypelem6 8833 ordtypelem7 8834 ordtypelem8 8835 ordtypelem9 8836 ordtypelem10 8837 |
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