Theorem hsmexlem1 10336

Description: Lemma for hsmex 10342. 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 9434	. . 3 ⊢ Ord dom 𝑂
3		relwdom 9471	. . . . . . . 8 ⊢ Rel ≼*
4	3	brrelex1i 5680	. . . . . . 7 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
5	4	adantl 481	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ∈ V)
6		uniexg 7685	. . . . . 6 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
7		sucexg 7750	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → suc ∪ 𝐴 ∈ V)
8	5, 6, 7	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → suc ∪ 𝐴 ∈ V)
9	1	oif 9435	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶𝐴
10		onsucuni 7770	. . . . . . . 8 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ⊆ suc ∪ 𝐴)
12		fss 6678	. . . . . . 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 9445	. . . . . . . 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 7724	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
18	17	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord ∪ 𝐴)
19		ordsuc 7756	. . . . . . 7 ⊢ (Ord ∪ 𝐴 ↔ Ord suc ∪ 𝐴)
20	18, 19	sylib 218	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord suc ∪ 𝐴)
21		smocdmdom 8300	. . . . . 6 ⊢ ((𝑂:dom 𝑂⟶suc ∪ 𝐴 ∧ Smo 𝑂 ∧ Ord suc ∪ 𝐴) → dom 𝑂 ⊆ suc ∪ 𝐴)
22	13, 16, 20, 21	syl3anc 1373	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ⊆ suc ∪ 𝐴)
23	8, 22	ssexd 5269	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ V)
24		elong 6325	. . . 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 9059	. . . 4 ⊢ (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28		sdomdom 8917	. . . 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 7720	. . . . . . . . . . 11 ⊢ E We On
32		wess 5610	. . . . . . . . . . 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 5606	. . . . . . . . . 10 ⊢ E Se 𝐴
35	1	oiiso2 9436	. . . . . . . . . 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 7269	. . . . . . . . 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 6771	. . . . . . . 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 8905	. . . . . . 7 ⊢ ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂–1-1-onto→𝐴) → dom 𝑂 ≈ 𝐴)
43	26, 5, 41, 42	syl3anc 1373	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≈ 𝐴)
44		endom 8916	. . . . . 6 ⊢ (dom 𝑂 ≈ 𝐴 → dom 𝑂 ≼ 𝐴)
45		domwdom 9479	. . . . . 6 ⊢ (dom 𝑂 ≼ 𝐴 → dom 𝑂 ≼* 𝐴)
46	43, 44, 45	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐴)
47		wdomtr 9480	. . . . 5 ⊢ ((dom 𝑂 ≼* 𝐴 ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
48	46, 47	sylancom 588	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
49		wdompwdom 9483	. . . 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 8944	. . 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 9466	. 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 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ⊆ wss 3901  𝒫 cpw 4554  ∪ cuni 4863   class class class wbr 5098   E cep 5523   Se wse 5575   We wwe 5576  dom cdm 5624  ran crn 5625  Ord word 6316  Oncon0 6317  suc csuc 6319  ⟶wf 6488  –1-1-onto→wf1o 6491  ‘cfv 6492   Isom wiso 6493  Smo wsmo 8277   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882  OrdIsocoi 9414  harchar 9461   ≼^* cwdom 9469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-isom 6501  df-riota 7315  df-ov 7361  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-smo 8278  df-recs 8303  df-en 8884  df-dom 8885  df-sdom 8886  df-oi 9415  df-har 9462  df-wdom 9470

This theorem is referenced by:  hsmexlem2  10337  hsmexlem4  10339