Theorem hsmexlem1 10182

Description: Lemma for hsmex 10188. 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 9288	. . 3 ⊢ Ord dom 𝑂
3		relwdom 9325	. . . . . . . 8 ⊢ Rel ≼*
4	3	brrelex1i 5643	. . . . . . 7 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
5	4	adantl 482	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ∈ V)
6		uniexg 7593	. . . . . 6 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
7		sucexg 7655	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → suc ∪ 𝐴 ∈ V)
8	5, 6, 7	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → suc ∪ 𝐴 ∈ V)
9	1	oif 9289	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶𝐴
10		onsucuni 7675	. . . . . . . 8 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ⊆ suc ∪ 𝐴)
12		fss 6617	. . . . . . 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 9299	. . . . . . . 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 7629	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
18	17	adantr 481	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord ∪ 𝐴)
19		ordsuc 7661	. . . . . . 7 ⊢ (Ord ∪ 𝐴 ↔ Ord suc ∪ 𝐴)
20	18, 19	sylib 217	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord suc ∪ 𝐴)
21		smorndom 8199	. . . . . 6 ⊢ ((𝑂:dom 𝑂⟶suc ∪ 𝐴 ∧ Smo 𝑂 ∧ Ord suc ∪ 𝐴) → dom 𝑂 ⊆ suc ∪ 𝐴)
22	13, 16, 20, 21	syl3anc 1370	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ⊆ suc ∪ 𝐴)
23	8, 22	ssexd 5248	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ V)
24		elong 6274	. . . 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 257	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ On)
27		canth2g 8918	. . . 4 ⊢ (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28		sdomdom 8768	. . . 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 7625	. . . . . . . . . . 11 ⊢ E We On
32		wess 5576	. . . . . . . . . . 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 5572	. . . . . . . . . 10 ⊢ E Se 𝐴
35	1	oiiso2 9290	. . . . . . . . . 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 7194	. . . . . . . . 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 6713	. . . . . . . 8 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → (𝑂:dom 𝑂–1-1-onto→ran 𝑂 ↔ 𝑂:dom 𝑂–1-1-onto→𝐴))
41	38, 40	mpbid 231	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂:dom 𝑂–1-1-onto→𝐴)
42		f1oen2g 8756	. . . . . . 7 ⊢ ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂–1-1-onto→𝐴) → dom 𝑂 ≈ 𝐴)
43	26, 5, 41, 42	syl3anc 1370	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≈ 𝐴)
44		endom 8767	. . . . . 6 ⊢ (dom 𝑂 ≈ 𝐴 → dom 𝑂 ≼ 𝐴)
45		domwdom 9333	. . . . . 6 ⊢ (dom 𝑂 ≼ 𝐴 → dom 𝑂 ≼* 𝐴)
46	43, 44, 45	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐴)
47		wdomtr 9334	. . . . 5 ⊢ ((dom 𝑂 ≼* 𝐴 ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
48	46, 47	sylancom 588	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
49		wdompwdom 9337	. . . 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 8793	. . 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 9320	. 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 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074   E cep 5494   Se wse 5542   We wwe 5543  dom cdm 5589  ran crn 5590  Ord word 6265  Oncon0 6266  suc csuc 6268  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433   Isom wiso 6434  Smo wsmo 8176   ≈ cen 8730   ≼ cdom 8731   ≺ csdm 8732  OrdIsocoi 9268  harchar 9315   ≼^* cwdom 9323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-smo 8177  df-recs 8202  df-en 8734  df-dom 8735  df-sdom 8736  df-oi 9269  df-har 9316  df-wdom 9324

This theorem is referenced by:  hsmexlem2  10183  hsmexlem4  10185