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Theorem hsmexlem1 10403
Description: Lemma for hsmex 10409. Bound the order type of a limited-cardinality set of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypothesis
Ref Expression
hsmexlem.o 𝑂 = OrdIso( E , 𝐴)
Assertion
Ref Expression
hsmexlem1 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))

Proof of Theorem hsmexlem1
StepHypRef Expression
1 hsmexlem.o . . . 4 𝑂 = OrdIso( E , 𝐴)
21oicl 9506 . . 3 Ord dom 𝑂
3 relwdom 9543 . . . . . . . 8 Rel ≼*
43brrelex1i 5724 . . . . . . 7 (𝐴* 𝐵𝐴 ∈ V)
54adantl 482 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ∈ V)
6 uniexg 7713 . . . . . 6 (𝐴 ∈ V → 𝐴 ∈ V)
7 sucexg 7776 . . . . . 6 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
85, 6, 73syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → suc 𝐴 ∈ V)
91oif 9507 . . . . . . 7 𝑂:dom 𝑂𝐴
10 onsucuni 7799 . . . . . . . 8 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
1110adantr 481 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ suc 𝐴)
12 fss 6721 . . . . . . 7 ((𝑂:dom 𝑂𝐴𝐴 ⊆ suc 𝐴) → 𝑂:dom 𝑂⟶suc 𝐴)
139, 11, 12sylancr 587 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂⟶suc 𝐴)
141oismo 9517 . . . . . . . 8 (𝐴 ⊆ On → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1514adantr 481 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1615simpld 495 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Smo 𝑂)
17 ssorduni 7749 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
1817adantr 481 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord 𝐴)
19 ordsuc 7784 . . . . . . 7 (Ord 𝐴 ↔ Ord suc 𝐴)
2018, 19sylib 217 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord suc 𝐴)
21 smocdmdom 8350 . . . . . 6 ((𝑂:dom 𝑂⟶suc 𝐴 ∧ Smo 𝑂 ∧ Ord suc 𝐴) → dom 𝑂 ⊆ suc 𝐴)
2213, 16, 20, 21syl3anc 1371 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ⊆ suc 𝐴)
238, 22ssexd 5317 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ V)
24 elong 6361 . . . 4 (dom 𝑂 ∈ V → (dom 𝑂 ∈ On ↔ Ord dom 𝑂))
2523, 24syl 17 . . 3 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (dom 𝑂 ∈ On ↔ Ord dom 𝑂))
262, 25mpbiri 257 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ On)
27 canth2g 9114 . . . 4 (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28 sdomdom 8959 . . . 4 (dom 𝑂 ≺ 𝒫 dom 𝑂 → dom 𝑂 ≼ 𝒫 dom 𝑂)
2923, 27, 283syl 18 . . 3 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ≼ 𝒫 dom 𝑂)
30 simpl 483 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ On)
31 epweon 7745 . . . . . . . . . . 11 E We On
32 wess 5656 . . . . . . . . . . 11 (𝐴 ⊆ On → ( E We On → E We 𝐴))
3330, 31, 32mpisyl 21 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → E We 𝐴)
34 epse 5652 . . . . . . . . . 10 E Se 𝐴
351oiiso2 9508 . . . . . . . . . 10 (( E We 𝐴 ∧ E Se 𝐴) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
3633, 34, 35sylancl 586 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
37 isof1o 7304 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂1-1-onto→ran 𝑂)
3836, 37syl 17 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂1-1-onto→ran 𝑂)
3915simprd 496 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → ran 𝑂 = 𝐴)
4039f1oeq3d 6817 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂1-1-onto𝐴))
4138, 40mpbid 231 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂1-1-onto𝐴)
42 f1oen2g 8947 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂1-1-onto𝐴) → dom 𝑂𝐴)
4326, 5, 41, 42syl3anc 1371 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂𝐴)
44 endom 8958 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂𝐴)
45 domwdom 9551 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂* 𝐴)
4643, 44, 453syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐴)
47 wdomtr 9552 . . . . 5 ((dom 𝑂* 𝐴𝐴* 𝐵) → dom 𝑂* 𝐵)
4846, 47sylancom 588 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐵)
49 wdompwdom 9555 . . . 4 (dom 𝑂* 𝐵 → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
5048, 49syl 17 . . 3 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
51 domtr 8986 . . 3 ((dom 𝑂 ≼ 𝒫 dom 𝑂 ∧ 𝒫 dom 𝑂 ≼ 𝒫 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
5229, 50, 51syl2anc 584 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
53 elharval 9538 . 2 (dom 𝑂 ∈ (har‘𝒫 𝐵) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐵))
5426, 52, 53sylanbrc 583 1 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3473  wss 3944  𝒫 cpw 4596   cuni 4901   class class class wbr 5141   E cep 5572   Se wse 5622   We wwe 5623  dom cdm 5669  ran crn 5670  Ord word 6352  Oncon0 6353  suc csuc 6355  wf 6528  1-1-ontowf1o 6531  cfv 6532   Isom wiso 6533  Smo wsmo 8327  cen 8919  cdom 8920  csdm 8921  OrdIsocoi 9486  harchar 9533  * cwdom 9541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-smo 8328  df-recs 8353  df-en 8923  df-dom 8924  df-sdom 8925  df-oi 9487  df-har 9534  df-wdom 9542
This theorem is referenced by:  hsmexlem2  10404  hsmexlem4  10406
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