Theorem hsmexlem1 10495

Description: Lemma for hsmex 10501. 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

Step	Hyp	Ref	Expression
1		hsmexlem.o	. . . 4 ⊢ 𝑂 = OrdIso( E , 𝐴)
2	1	oicl 9598	. . 3 ⊢ Ord dom 𝑂
3		relwdom 9635	. . . . . . . 8 ⊢ Rel ≼*
4	3	brrelex1i 5756	. . . . . . 7 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
5	4	adantl 481	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ∈ V)
6		uniexg 7775	. . . . . 6 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
7		sucexg 7841	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → suc ∪ 𝐴 ∈ V)
8	5, 6, 7	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → suc ∪ 𝐴 ∈ V)
9	1	oif 9599	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶𝐴
10		onsucuni 7864	. . . . . . . 8 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ⊆ suc ∪ 𝐴)
12		fss 6763	. . . . . . 7 ⊢ ((𝑂:dom 𝑂⟶𝐴 ∧ 𝐴 ⊆ suc ∪ 𝐴) → 𝑂:dom 𝑂⟶suc ∪ 𝐴)
13	9, 11, 12	sylancr 586	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂:dom 𝑂⟶suc ∪ 𝐴)
14	1	oismo 9609	. . . . . . . 8 ⊢ (𝐴 ⊆ On → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
15	14	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
16	15	simpld 494	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Smo 𝑂)
17		ssorduni 7814	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
18	17	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord ∪ 𝐴)
19		ordsuc 7849	. . . . . . 7 ⊢ (Ord ∪ 𝐴 ↔ Ord suc ∪ 𝐴)
20	18, 19	sylib 218	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord suc ∪ 𝐴)
21		smocdmdom 8424	. . . . . 6 ⊢ ((𝑂:dom 𝑂⟶suc ∪ 𝐴 ∧ Smo 𝑂 ∧ Ord suc ∪ 𝐴) → dom 𝑂 ⊆ suc ∪ 𝐴)
22	13, 16, 20, 21	syl3anc 1371	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ⊆ suc ∪ 𝐴)
23	8, 22	ssexd 5342	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ V)
24		elong 6403	. . . 4 ⊢ (dom 𝑂 ∈ V → (dom 𝑂 ∈ On ↔ Ord dom 𝑂))
25	23, 24	syl 17	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → (dom 𝑂 ∈ On ↔ Ord dom 𝑂))
26	2, 25	mpbiri 258	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ On)
27		canth2g 9197	. . . 4 ⊢ (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28		sdomdom 9040	. . . 4 ⊢ (dom 𝑂 ≺ 𝒫 dom 𝑂 → dom 𝑂 ≼ 𝒫 dom 𝑂)
29	23, 27, 28	3syl 18	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼ 𝒫 dom 𝑂)
30		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ⊆ On)
31		epweon 7810	. . . . . . . . . . 11 ⊢ E We On
32		wess 5686	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴))
33	30, 31, 32	mpisyl 21	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → E We 𝐴)
34		epse 5682	. . . . . . . . . 10 ⊢ E Se 𝐴
35	1	oiiso2 9600	. . . . . . . . . 10 ⊢ (( E We 𝐴 ∧ E Se 𝐴) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
36	33, 34, 35	sylancl 585	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
37		isof1o 7359	. . . . . . . . 9 ⊢ (𝑂 Isom E , E (dom 𝑂, ran 𝑂) → 𝑂:dom 𝑂–1-1-onto→ran 𝑂)
38	36, 37	syl 17	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂:dom 𝑂–1-1-onto→ran 𝑂)
39	15	simprd 495	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → ran 𝑂 = 𝐴)
40	39	f1oeq3d 6859	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → (𝑂:dom 𝑂–1-1-onto→ran 𝑂 ↔ 𝑂:dom 𝑂–1-1-onto→𝐴))
41	38, 40	mpbid 232	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂:dom 𝑂–1-1-onto→𝐴)
42		f1oen2g 9028	. . . . . . 7 ⊢ ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂–1-1-onto→𝐴) → dom 𝑂 ≈ 𝐴)
43	26, 5, 41, 42	syl3anc 1371	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≈ 𝐴)
44		endom 9039	. . . . . 6 ⊢ (dom 𝑂 ≈ 𝐴 → dom 𝑂 ≼ 𝐴)
45		domwdom 9643	. . . . . 6 ⊢ (dom 𝑂 ≼ 𝐴 → dom 𝑂 ≼* 𝐴)
46	43, 44, 45	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐴)
47		wdomtr 9644	. . . . 5 ⊢ ((dom 𝑂 ≼* 𝐴 ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
48	46, 47	sylancom 587	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
49		wdompwdom 9647	. . . 4 ⊢ (dom 𝑂 ≼* 𝐵 → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
50	48, 49	syl 17	. . 3 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝒫 dom 𝑂 ≼ 𝒫 𝐵)
51		domtr 9067	. . 3 ⊢ ((dom 𝑂 ≼ 𝒫 dom 𝑂 ∧ 𝒫 dom 𝑂 ≼ 𝒫 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
52	29, 50, 51	syl2anc 583	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
53		elharval 9630	. 2 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐵) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐵))
54	26, 52, 53	sylanbrc 582	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166   E cep 5598   Se wse 5650   We wwe 5651  dom cdm 5700  ran crn 5701  Ord word 6394  Oncon0 6395  suc csuc 6397  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573   Isom wiso 6574  Smo wsmo 8401   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002  OrdIsocoi 9578  harchar 9625   ≼^* cwdom 9633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-smo 8402  df-recs 8427  df-en 9004  df-dom 9005  df-sdom 9006  df-oi 9579  df-har 9626  df-wdom 9634

This theorem is referenced by:  hsmexlem2  10496  hsmexlem4  10498