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Theorem hsmexlem1 9886
 Description: Lemma for hsmex 9892. 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 9026 . . 3 Ord dom 𝑂
3 relwdom 9063 . . . . . . . 8 Rel ≼*
43brrelex1i 5577 . . . . . . 7 (𝐴* 𝐵𝐴 ∈ V)
54adantl 485 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ∈ V)
6 uniexg 7464 . . . . . 6 (𝐴 ∈ V → 𝐴 ∈ V)
7 sucexg 7524 . . . . . 6 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
85, 6, 73syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → suc 𝐴 ∈ V)
91oif 9027 . . . . . . 7 𝑂:dom 𝑂𝐴
10 onsucuni 7542 . . . . . . . 8 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
1110adantr 484 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ suc 𝐴)
12 fss 6512 . . . . . . 7 ((𝑂:dom 𝑂𝐴𝐴 ⊆ suc 𝐴) → 𝑂:dom 𝑂⟶suc 𝐴)
139, 11, 12sylancr 590 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂⟶suc 𝐴)
141oismo 9037 . . . . . . . 8 (𝐴 ⊆ On → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1514adantr 484 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1615simpld 498 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Smo 𝑂)
17 ssorduni 7499 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
1817adantr 484 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord 𝐴)
19 ordsuc 7528 . . . . . . 7 (Ord 𝐴 ↔ Ord suc 𝐴)
2018, 19sylib 221 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord suc 𝐴)
21 smorndom 8015 . . . . . 6 ((𝑂:dom 𝑂⟶suc 𝐴 ∧ Smo 𝑂 ∧ Ord suc 𝐴) → dom 𝑂 ⊆ suc 𝐴)
2213, 16, 20, 21syl3anc 1368 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ⊆ suc 𝐴)
238, 22ssexd 5194 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ V)
24 elong 6177 . . . 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 261 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ On)
27 canth2g 8693 . . . 4 (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28 sdomdom 8555 . . . 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 486 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ On)
31 epweon 7496 . . . . . . . . . . 11 E We On
32 wess 5511 . . . . . . . . . . 11 (𝐴 ⊆ On → ( E We On → E We 𝐴))
3330, 31, 32mpisyl 21 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → E We 𝐴)
34 epse 5507 . . . . . . . . . 10 E Se 𝐴
351oiiso2 9028 . . . . . . . . . 10 (( E We 𝐴 ∧ E Se 𝐴) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
3633, 34, 35sylancl 589 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
37 isof1o 7070 . . . . . . . . 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 499 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → ran 𝑂 = 𝐴)
4039f1oeq3d 6599 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂1-1-onto𝐴))
4138, 40mpbid 235 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂1-1-onto𝐴)
42 f1oen2g 8544 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂1-1-onto𝐴) → dom 𝑂𝐴)
4326, 5, 41, 42syl3anc 1368 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂𝐴)
44 endom 8554 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂𝐴)
45 domwdom 9071 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂* 𝐴)
4643, 44, 453syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐴)
47 wdomtr 9072 . . . . 5 ((dom 𝑂* 𝐴𝐴* 𝐵) → dom 𝑂* 𝐵)
4846, 47sylancom 591 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐵)
49 wdompwdom 9075 . . . 4 (dom 𝑂* 𝐵 → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
5048, 49syl 17 . . 3 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
51 domtr 8580 . . 3 ((dom 𝑂 ≼ 𝒫 dom 𝑂 ∧ 𝒫 dom 𝑂 ≼ 𝒫 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
5229, 50, 51syl2anc 587 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
53 elharval 9058 . 2 (dom 𝑂 ∈ (har‘𝒫 𝐵) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐵))
5426, 52, 53sylanbrc 586 1 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  𝒫 cpw 4494  ∪ cuni 4798   class class class wbr 5032   E cep 5434   Se wse 5481   We wwe 5482  dom cdm 5524  ran crn 5525  Ord word 6168  Oncon0 6169  suc csuc 6171  ⟶wf 6331  –1-1-onto→wf1o 6334  ‘cfv 6335   Isom wiso 6336  Smo wsmo 7992   ≈ cen 8524   ≼ cdom 8525   ≺ csdm 8526  OrdIsocoi 9006  harchar 9053   ≼* cwdom 9061 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-se 5484  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-isom 6344  df-riota 7108  df-wrecs 7957  df-smo 7993  df-recs 8018  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-oi 9007  df-har 9054  df-wdom 9062 This theorem is referenced by:  hsmexlem2  9887  hsmexlem4  9889
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