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Mirrors > Home > MPE Home > Th. List > ordtypelem4 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 8990. (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 8024 | . . . . . . 7 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
3 | 2 | simpli 486 | . . . . . 6 ⊢ Fun 𝐹 |
4 | funres 6391 | . . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑇)) | |
5 | 3, 4 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑇)) |
6 | 5 | funfnd 6380 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇)) |
7 | dmres 5869 | . . . . 5 ⊢ dom (𝐹 ↾ 𝑇) = (𝑇 ∩ dom 𝐹) | |
8 | 7 | fneq2i 6445 | . . . 4 ⊢ ((𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇) ↔ (𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹)) |
9 | 6, 8 | sylib 220 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹)) |
10 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹)) | |
11 | 10 | elin1d 4174 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ 𝑇) |
12 | 11 | fvresd 6684 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) = (𝐹‘𝑎)) |
13 | ssrab2 4055 | . . . . . . 7 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} | |
14 | ssrab2 4055 | . . . . . . 7 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ⊆ 𝐴 | |
15 | 13, 14 | sstri 3975 | . . . . . 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 8978 | . . . . . 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 3964 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑎) ∈ 𝐴) |
24 | 12, 23 | eqeltrd 2913 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴) |
25 | 24 | ralrimiva 3182 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴) |
26 | ffnfv 6876 | . . 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 585 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴) |
28 | 1, 16, 17, 18, 19, 20, 21 | ordtypelem1 8976 | . . 3 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) |
29 | 28 | feq1d 6493 | . 2 ⊢ (𝜑 → (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴)) |
30 | 27, 29 | mpbird 259 | 1 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 {crab 3142 Vcvv 3494 ∩ cin 3934 class class class wbr 5058 ↦ cmpt 5138 Se wse 5506 We wwe 5507 dom cdm 5549 ran crn 5550 ↾ cres 5551 “ cima 5552 Oncon0 6185 Lim wlim 6186 Fun wfun 6343 Fn wfn 6344 ⟶wf 6345 ‘cfv 6349 ℩crio 7107 recscrecs 8001 OrdIsocoi 8967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-wrecs 7941 df-recs 8002 df-oi 8968 |
This theorem is referenced by: ordtypelem5 8980 ordtypelem6 8981 ordtypelem7 8982 ordtypelem8 8983 ordtypelem9 8984 ordtypelem10 8985 |
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