![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ordtypelem4 | Structured version Visualization version GIF version |
Description: Lemma for ordtype 9527. (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 8394 | . . . . . . 7 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
3 | 2 | simpli 485 | . . . . . 6 ⊢ Fun 𝐹 |
4 | funres 6591 | . . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑇)) | |
5 | 3, 4 | mp1i 13 | . . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑇)) |
6 | 5 | funfnd 6580 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑇) Fn dom (𝐹 ↾ 𝑇)) |
7 | dmres 6004 | . . . . 5 ⊢ dom (𝐹 ↾ 𝑇) = (𝑇 ∩ dom 𝐹) | |
8 | 7 | fneq2i 6648 | . . . 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 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ (𝑇 ∩ dom 𝐹)) | |
11 | 10 | elin1d 4199 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → 𝑎 ∈ 𝑇) |
12 | 11 | fvresd 6912 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) = (𝐹‘𝑎)) |
13 | ssrab2 4078 | . . . . . . 7 ⊢ {𝑣 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ∣ ∀𝑢 ∈ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ¬ 𝑢𝑅𝑣} ⊆ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} | |
14 | ssrab2 4078 | . . . . . . 7 ⊢ {𝑤 ∈ 𝐴 ∣ ∀𝑗 ∈ (𝐹 “ 𝑎)𝑗𝑅𝑤} ⊆ 𝐴 | |
15 | 13, 14 | sstri 3992 | . . . . . 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 9515 | . . . . . 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 3981 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → (𝐹‘𝑎) ∈ 𝐴) |
24 | 12, 23 | eqeltrd 2834 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑇 ∩ dom 𝐹)) → ((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴) |
25 | 24 | ralrimiva 3147 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ (𝑇 ∩ dom 𝐹)((𝐹 ↾ 𝑇)‘𝑎) ∈ 𝐴) |
26 | ffnfv 7118 | . . 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 584 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴) |
28 | 1, 16, 17, 18, 19, 20, 21 | ordtypelem1 9513 | . . 3 ⊢ (𝜑 → 𝑂 = (𝐹 ↾ 𝑇)) |
29 | 28 | feq1d 6703 | . 2 ⊢ (𝜑 → (𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴 ↔ (𝐹 ↾ 𝑇):(𝑇 ∩ dom 𝐹)⟶𝐴)) |
30 | 27, 29 | mpbird 257 | 1 ⊢ (𝜑 → 𝑂:(𝑇 ∩ dom 𝐹)⟶𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ∩ cin 3948 class class class wbr 5149 ↦ cmpt 5232 Se wse 5630 We wwe 5631 dom cdm 5677 ran crn 5678 ↾ cres 5679 “ cima 5680 Oncon0 6365 Lim wlim 6366 Fun wfun 6538 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 ℩crio 7364 recscrecs 8370 OrdIsocoi 9504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-oi 9505 |
This theorem is referenced by: ordtypelem5 9517 ordtypelem6 9518 ordtypelem7 9519 ordtypelem8 9520 ordtypelem9 9521 ordtypelem10 9522 |
Copyright terms: Public domain | W3C validator |