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Theorem hsmexlem1 10463
Description: Lemma for hsmex 10469. 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 9566 . . 3 Ord dom 𝑂
3 relwdom 9603 . . . . . . . 8 Rel ≼*
43brrelex1i 5744 . . . . . . 7 (𝐴* 𝐵𝐴 ∈ V)
54adantl 481 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ∈ V)
6 uniexg 7758 . . . . . 6 (𝐴 ∈ V → 𝐴 ∈ V)
7 sucexg 7824 . . . . . 6 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
85, 6, 73syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → suc 𝐴 ∈ V)
91oif 9567 . . . . . . 7 𝑂:dom 𝑂𝐴
10 onsucuni 7847 . . . . . . . 8 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
1110adantr 480 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ suc 𝐴)
12 fss 6752 . . . . . . 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 9577 . . . . . . . 8 (𝐴 ⊆ On → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1514adantr 480 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1615simpld 494 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Smo 𝑂)
17 ssorduni 7797 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
1817adantr 480 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord 𝐴)
19 ordsuc 7832 . . . . . . 7 (Ord 𝐴 ↔ Ord suc 𝐴)
2018, 19sylib 218 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord suc 𝐴)
21 smocdmdom 8406 . . . . . 6 ((𝑂:dom 𝑂⟶suc 𝐴 ∧ Smo 𝑂 ∧ Ord suc 𝐴) → dom 𝑂 ⊆ suc 𝐴)
2213, 16, 20, 21syl3anc 1370 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ⊆ suc 𝐴)
238, 22ssexd 5329 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ V)
24 elong 6393 . . . 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 258 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ On)
27 canth2g 9169 . . . 4 (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28 sdomdom 9018 . . . 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 482 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ On)
31 epweon 7793 . . . . . . . . . . 11 E We On
32 wess 5674 . . . . . . . . . . 11 (𝐴 ⊆ On → ( E We On → E We 𝐴))
3330, 31, 32mpisyl 21 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → E We 𝐴)
34 epse 5670 . . . . . . . . . 10 E Se 𝐴
351oiiso2 9568 . . . . . . . . . 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 7342 . . . . . . . . 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 495 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → ran 𝑂 = 𝐴)
4039f1oeq3d 6845 . . . . . . . 8 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (𝑂:dom 𝑂1-1-onto→ran 𝑂𝑂:dom 𝑂1-1-onto𝐴))
4138, 40mpbid 232 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂1-1-onto𝐴)
42 f1oen2g 9007 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂1-1-onto𝐴) → dom 𝑂𝐴)
4326, 5, 41, 42syl3anc 1370 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂𝐴)
44 endom 9017 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂𝐴)
45 domwdom 9611 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂* 𝐴)
4643, 44, 453syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐴)
47 wdomtr 9612 . . . . 5 ((dom 𝑂* 𝐴𝐴* 𝐵) → dom 𝑂* 𝐵)
4846, 47sylancom 588 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐵)
49 wdompwdom 9615 . . . 4 (dom 𝑂* 𝐵 → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
5048, 49syl 17 . . 3 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
51 domtr 9045 . . 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 9598 . 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 206  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  wss 3962  𝒫 cpw 4604   cuni 4911   class class class wbr 5147   E cep 5587   Se wse 5638   We wwe 5639  dom cdm 5688  ran crn 5689  Ord word 6384  Oncon0 6385  suc csuc 6387  wf 6558  1-1-ontowf1o 6561  cfv 6562   Isom wiso 6563  Smo wsmo 8383  cen 8980  cdom 8981  csdm 8982  OrdIsocoi 9546  harchar 9593  * cwdom 9601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-smo 8384  df-recs 8409  df-en 8984  df-dom 8985  df-sdom 8986  df-oi 9547  df-har 9594  df-wdom 9602
This theorem is referenced by:  hsmexlem2  10464  hsmexlem4  10466
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