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Mirrors > Home > MPE Home > Th. List > ordtypelem4 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 9529. (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 8396 | . . . . . . 7 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
3 | 2 | simpli 482 | . . . . . 6 ⊢ Fun 𝐹 |
4 | funres 6589 | . . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑇)) | |
5 | 3, 4 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑇)) |
6 | 5 | funfnd 6578 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇)) |
7 | dmres 6002 | . . . . 5 ⊢ dom (𝐹 ↾ 𝑇) = (𝑇 ∩ dom 𝐹) | |
8 | 7 | fneq2i 6646 | . . . 4 ⊢ ((𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇) ↔ (𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹)) |
9 | 6, 8 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn (𝑇 ∩ dom 𝐹)) |
10 | simpr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹)) | |
11 | 10 | elin1d 4197 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ 𝑇) |
12 | 11 | fvresd 6910 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) = (𝐹‘𝑎)) |
13 | ssrab2 4076 | . . . . . . 7 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} | |
14 | ssrab2 4076 | . . . . . . 7 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ⊆ 𝐴 | |
15 | 13, 14 | sstri 3990 | . . . . . 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 9517 | . . . . . 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 | sselid 3979 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑎) ∈ 𝐴) |
24 | 12, 23 | eqeltrd 2831 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴) |
25 | 24 | ralrimiva 3144 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴) |
26 | ffnfv 7119 | . . 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 581 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴) |
28 | 1, 16, 17, 18, 19, 20, 21 | ordtypelem1 9515 | . . 3 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) |
29 | 28 | feq1d 6701 | . 2 ⊢ (𝜑 → (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴)) |
30 | 27, 29 | mpbird 256 | 1 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 {crab 3430 Vcvv 3472 ∩ cin 3946 class class class wbr 5147 ↦ cmpt 5230 Se wse 5628 We wwe 5629 dom cdm 5675 ran crn 5676 ↾ cres 5677 “ cima 5678 Oncon0 6363 Lim wlim 6364 Fun wfun 6536 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 ℩crio 7366 recscrecs 8372 OrdIsocoi 9506 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7367 df-ov 7414 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-oi 9507 |
This theorem is referenced by: ordtypelem5 9519 ordtypelem6 9520 ordtypelem7 9521 ordtypelem8 9522 ordtypelem9 9523 ordtypelem10 9524 |
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