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Theorem hsmexlem1 9193
Description: Lemma for hsmex 9199. 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 8379 . . 3 Ord dom 𝑂
3 relwdom 8416 . . . . . . . 8 Rel ≼*
43brrelexi 5123 . . . . . . 7 (𝐴* 𝐵𝐴 ∈ V)
54adantl 482 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ∈ V)
6 uniexg 6909 . . . . . 6 (𝐴 ∈ V → 𝐴 ∈ V)
7 sucexg 6958 . . . . . 6 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
85, 6, 73syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → suc 𝐴 ∈ V)
91oif 8380 . . . . . . 7 𝑂:dom 𝑂𝐴
10 onsucuni 6976 . . . . . . . 8 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
1110adantr 481 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ suc 𝐴)
12 fss 6015 . . . . . . 7 ((𝑂:dom 𝑂𝐴𝐴 ⊆ suc 𝐴) → 𝑂:dom 𝑂⟶suc 𝐴)
139, 11, 12sylancr 694 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂⟶suc 𝐴)
141oismo 8390 . . . . . . . 8 (𝐴 ⊆ On → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1514adantr 481 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1615simpld 475 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Smo 𝑂)
17 ssorduni 6933 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
1817adantr 481 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord 𝐴)
19 ordsuc 6962 . . . . . . 7 (Ord 𝐴 ↔ Ord suc 𝐴)
2018, 19sylib 208 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord suc 𝐴)
21 smorndom 7411 . . . . . 6 ((𝑂:dom 𝑂⟶suc 𝐴 ∧ Smo 𝑂 ∧ Ord suc 𝐴) → dom 𝑂 ⊆ suc 𝐴)
2213, 16, 20, 21syl3anc 1323 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ⊆ suc 𝐴)
238, 22ssexd 4770 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ V)
24 elong 5693 . . . 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 248 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ On)
27 canth2g 8059 . . . 4 (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28 sdomdom 7928 . . . 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 473 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ On)
31 epweon 6931 . . . . . . . . . . 11 E We On
32 wess 5066 . . . . . . . . . . 11 (𝐴 ⊆ On → ( E We On → E We 𝐴))
3330, 31, 32mpisyl 21 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → E We 𝐴)
34 epse 5062 . . . . . . . . . 10 E Se 𝐴
351oiiso2 8381 . . . . . . . . . 10 (( E We 𝐴 ∧ E Se 𝐴) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
3633, 34, 35sylancl 693 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
37 isof1o 6528 . . . . . . . . 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 479 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → ran 𝑂 = 𝐴)
40 f1oeq3 6088 . . . . . . . . 9 (ran 𝑂 = 𝐴 → (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂1-1-onto𝐴))
4139, 40syl 17 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂1-1-onto𝐴))
4238, 41mpbid 222 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂1-1-onto𝐴)
43 f1oen2g 7917 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂1-1-onto𝐴) → dom 𝑂𝐴)
4426, 5, 42, 43syl3anc 1323 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂𝐴)
45 endom 7927 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂𝐴)
46 domwdom 8424 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂* 𝐴)
4744, 45, 463syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐴)
48 wdomtr 8425 . . . . 5 ((dom 𝑂* 𝐴𝐴* 𝐵) → dom 𝑂* 𝐵)
4947, 48sylancom 700 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐵)
50 wdompwdom 8428 . . . 4 (dom 𝑂* 𝐵 → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
5149, 50syl 17 . . 3 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
52 domtr 7954 . . 3 ((dom 𝑂 ≼ 𝒫 dom 𝑂 ∧ 𝒫 dom 𝑂 ≼ 𝒫 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
5329, 51, 52syl2anc 692 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
54 elharval 8413 . 2 (dom 𝑂 ∈ (har‘𝒫 𝐵) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐵))
5526, 53, 54sylanbrc 697 1 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  wss 3560  𝒫 cpw 4135   cuni 4407   class class class wbr 4618   E cep 4988   Se wse 5036   We wwe 5037  dom cdm 5079  ran crn 5080  Ord word 5684  Oncon0 5685  suc csuc 5687  wf 5846  1-1-ontowf1o 5849  cfv 5850   Isom wiso 5851  Smo wsmo 7388  cen 7897  cdom 7898  csdm 7899  OrdIsocoi 8359  harchar 8406  * cwdom 8407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-isom 5859  df-riota 6566  df-wrecs 7353  df-smo 7389  df-recs 7414  df-er 7688  df-en 7901  df-dom 7902  df-sdom 7903  df-oi 8360  df-har 8408  df-wdom 8409
This theorem is referenced by:  hsmexlem2  9194  hsmexlem4  9196
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