Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem1 Structured version   Visualization version   GIF version

Theorem hsmexlem1 9131
 Description: Lemma for hsmex 9137. 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 8317 . . 3 Ord dom 𝑂
3 relwdom 8354 . . . . . . . 8 Rel ≼*
43brrelexi 5082 . . . . . . 7 (𝐴* 𝐵𝐴 ∈ V)
54adantl 481 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ∈ V)
6 uniexg 6853 . . . . . 6 (𝐴 ∈ V → 𝐴 ∈ V)
7 sucexg 6902 . . . . . 6 ( 𝐴 ∈ V → suc 𝐴 ∈ V)
85, 6, 73syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → suc 𝐴 ∈ V)
91oif 8318 . . . . . . 7 𝑂:dom 𝑂𝐴
10 onsucuni 6920 . . . . . . . 8 (𝐴 ⊆ On → 𝐴 ⊆ suc 𝐴)
1110adantr 480 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ suc 𝐴)
12 fss 5969 . . . . . . 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 8328 . . . . . . . 8 (𝐴 ⊆ On → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1514adantr 480 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
1615simpld 474 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Smo 𝑂)
17 ssorduni 6877 . . . . . . . 8 (𝐴 ⊆ On → Ord 𝐴)
1817adantr 480 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord 𝐴)
19 ordsuc 6906 . . . . . . 7 (Ord 𝐴 ↔ Ord suc 𝐴)
2018, 19sylib 207 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → Ord suc 𝐴)
21 smorndom 7352 . . . . . 6 ((𝑂:dom 𝑂⟶suc 𝐴 ∧ Smo 𝑂 ∧ Ord suc 𝐴) → dom 𝑂 ⊆ suc 𝐴)
2213, 16, 20, 21syl3anc 1318 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ⊆ suc 𝐴)
238, 22ssexd 4733 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ V)
24 elong 5648 . . . 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 247 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ On)
27 canth2g 7999 . . . 4 (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28 sdomdom 7869 . . . 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 472 . . . . . . . . . . 11 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝐴 ⊆ On)
31 epweon 6875 . . . . . . . . . . 11 E We On
32 wess 5025 . . . . . . . . . . 11 (𝐴 ⊆ On → ( E We On → E We 𝐴))
3330, 31, 32mpisyl 21 . . . . . . . . . 10 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → E We 𝐴)
34 epse 5021 . . . . . . . . . 10 E Se 𝐴
351oiiso2 8319 . . . . . . . . . 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 6473 . . . . . . . . 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 478 . . . . . . . . 9 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → ran 𝑂 = 𝐴)
40 f1oeq3 6042 . . . . . . . . 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 221 . . . . . . 7 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝑂:dom 𝑂1-1-onto𝐴)
43 f1oen2g 7858 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂1-1-onto𝐴) → dom 𝑂𝐴)
4426, 5, 42, 43syl3anc 1318 . . . . . 6 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂𝐴)
45 endom 7868 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂𝐴)
46 domwdom 8362 . . . . . 6 (dom 𝑂𝐴 → dom 𝑂* 𝐴)
4744, 45, 463syl 18 . . . . 5 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐴)
48 wdomtr 8363 . . . . 5 ((dom 𝑂* 𝐴𝐴* 𝐵) → dom 𝑂* 𝐵)
4947, 48sylancom 698 . . . 4 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂* 𝐵)
50 wdompwdom 8366 . . . 4 (dom 𝑂* 𝐵 → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
5149, 50syl 17 . . 3 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
52 domtr 7895 . . 3 ((dom 𝑂 ≼ 𝒫 dom 𝑂 ∧ 𝒫 dom 𝑂 ≼ 𝒫 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
5329, 51, 52syl2anc 691 . 2 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
54 elharval 8351 . 2 (dom 𝑂 ∈ (har‘𝒫 𝐵) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐵))
5526, 53, 54sylanbrc 695 1 ((𝐴 ⊆ On ∧ 𝐴* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583   E cep 4947   Se wse 4995   We wwe 4996  dom cdm 5038  ran crn 5039  Ord word 5639  Oncon0 5640  suc csuc 5642  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804   Isom wiso 5805  Smo wsmo 7329   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  OrdIsocoi 8297  harchar 8344   ≼* cwdom 8345 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-smo 7330  df-recs 7355  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-oi 8298  df-har 8346  df-wdom 8347 This theorem is referenced by:  hsmexlem2  9132  hsmexlem4  9134
 Copyright terms: Public domain W3C validator