Theorem hsmexlem1 9694

Description: Lemma for hsmex 9700. 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 8839	. . 3 ⊢ Ord dom 𝑂
3		relwdom 8876	. . . . . . . 8 ⊢ Rel ≼*
4	3	brrelex1i 5494	. . . . . . 7 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
5	4	adantl 482	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ∈ V)
6		uniexg 7325	. . . . . 6 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
7		sucexg 7381	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → suc ∪ 𝐴 ∈ V)
8	5, 6, 7	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → suc ∪ 𝐴 ∈ V)
9	1	oif 8840	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶𝐴
10		onsucuni 7399	. . . . . . . 8 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ⊆ suc ∪ 𝐴)
12		fss 6395	. . . . . . 7 ⊢ ((𝑂:dom 𝑂⟶𝐴 ∧ 𝐴 ⊆ suc ∪ 𝐴) → 𝑂:dom 𝑂⟶suc ∪ 𝐴)
13	9, 11, 12	sylancr 587	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂:dom 𝑂⟶suc ∪ 𝐴)
14	1	oismo 8850	. . . . . . . 8 ⊢ (𝐴 ⊆ On → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
15	14	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → (Smo 𝑂 ∧ ran 𝑂 = 𝐴))
16	15	simpld 495	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Smo 𝑂)
17		ssorduni 7356	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
18	17	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord ∪ 𝐴)
19		ordsuc 7385	. . . . . . 7 ⊢ (Ord ∪ 𝐴 ↔ Ord suc ∪ 𝐴)
20	18, 19	sylib 219	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord suc ∪ 𝐴)
21		smorndom 7857	. . . . . 6 ⊢ ((𝑂:dom 𝑂⟶suc ∪ 𝐴 ∧ Smo 𝑂 ∧ Ord suc ∪ 𝐴) → dom 𝑂 ⊆ suc ∪ 𝐴)
22	13, 16, 20, 21	syl3anc 1364	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ⊆ suc ∪ 𝐴)
23	8, 22	ssexd 5119	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ V)
24		elong 6074	. . . 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 259	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ On)
27		canth2g 8518	. . . 4 ⊢ (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28		sdomdom 8385	. . . 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 483	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ⊆ On)
31		epweon 7353	. . . . . . . . . . 11 ⊢ E We On
32		wess 5430	. . . . . . . . . . 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 5426	. . . . . . . . . 10 ⊢ E Se 𝐴
35	1	oiiso2 8841	. . . . . . . . . 10 ⊢ (( E We 𝐴 ∧ E Se 𝐴) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
36	33, 34, 35	sylancl 586	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
37		isof1o 6939	. . . . . . . . 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 496	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → ran 𝑂 = 𝐴)
40	39	f1oeq3d 6480	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → (𝑂:dom 𝑂–1-1-onto→ran 𝑂 ↔ 𝑂:dom 𝑂–1-1-onto→𝐴))
41	38, 40	mpbid 233	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂:dom 𝑂–1-1-onto→𝐴)
42		f1oen2g 8374	. . . . . . 7 ⊢ ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂–1-1-onto→𝐴) → dom 𝑂 ≈ 𝐴)
43	26, 5, 41, 42	syl3anc 1364	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≈ 𝐴)
44		endom 8384	. . . . . 6 ⊢ (dom 𝑂 ≈ 𝐴 → dom 𝑂 ≼ 𝐴)
45		domwdom 8884	. . . . . 6 ⊢ (dom 𝑂 ≼ 𝐴 → dom 𝑂 ≼* 𝐴)
46	43, 44, 45	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐴)
47		wdomtr 8885	. . . . 5 ⊢ ((dom 𝑂 ≼* 𝐴 ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
48	46, 47	sylancom 588	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
49		wdompwdom 8888	. . . 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 8410	. . 3 ⊢ ((dom 𝑂 ≼ 𝒫 dom 𝑂 ∧ 𝒫 dom 𝑂 ≼ 𝒫 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
52	29, 50, 51	syl2anc 584	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
53		elharval 8873	. 2 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐵) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐵))
54	26, 52, 53	sylanbrc 583	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1522   ∈ wcel 2081  Vcvv 3437   ⊆ wss 3859  𝒫 cpw 4453  ∪ cuni 4745   class class class wbr 4962   E cep 5352   Se wse 5400   We wwe 5401  dom cdm 5443  ran crn 5444  Ord word 6065  Oncon0 6066  suc csuc 6068  ⟶wf 6221  –1-1-onto→wf1o 6224  ‘cfv 6225   Isom wiso 6226  Smo wsmo 7834   ≈ cen 8354   ≼ cdom 8355   ≺ csdm 8356  OrdIsocoi 8819  harchar 8866   ≼^* cwdom 8867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-se 5403  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-isom 6234  df-riota 6977  df-wrecs 7798  df-smo 7835  df-recs 7860  df-er 8139  df-en 8358  df-dom 8359  df-sdom 8360  df-oi 8820  df-har 8868  df-wdom 8869

This theorem is referenced by:  hsmexlem2  9695  hsmexlem4  9697