Theorem hsmexlem1 10342

Description: Lemma for hsmex 10348. 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 9438	. . 3 ⊢ Ord dom 𝑂
3		relwdom 9475	. . . . . . . 8 ⊢ Rel ≼*
4	3	brrelex1i 5681	. . . . . . 7 ⊢ (𝐴 ≼* 𝐵 → 𝐴 ∈ V)
5	4	adantl 481	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ∈ V)
6		uniexg 7688	. . . . . 6 ⊢ (𝐴 ∈ V → ∪ 𝐴 ∈ V)
7		sucexg 7753	. . . . . 6 ⊢ (∪ 𝐴 ∈ V → suc ∪ 𝐴 ∈ V)
8	5, 6, 7	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → suc ∪ 𝐴 ∈ V)
9	1	oif 9439	. . . . . . 7 ⊢ 𝑂:dom 𝑂⟶𝐴
10		onsucuni 7773	. . . . . . . 8 ⊢ (𝐴 ⊆ On → 𝐴 ⊆ suc ∪ 𝐴)
11	10	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝐴 ⊆ suc ∪ 𝐴)
12		fss 6679	. . . . . . 7 ⊢ ((𝑂:dom 𝑂⟶𝐴 ∧ 𝐴 ⊆ suc ∪ 𝐴) → 𝑂:dom 𝑂⟶suc ∪ 𝐴)
13	9, 11, 12	sylancr 588	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂:dom 𝑂⟶suc ∪ 𝐴)
14	1	oismo 9449	. . . . . . . 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 7727	. . . . . . . 8 ⊢ (𝐴 ⊆ On → Ord ∪ 𝐴)
18	17	adantr 480	. . . . . . 7 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord ∪ 𝐴)
19		ordsuc 7759	. . . . . . 7 ⊢ (Ord ∪ 𝐴 ↔ Ord suc ∪ 𝐴)
20	18, 19	sylib 218	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → Ord suc ∪ 𝐴)
21		smocdmdom 8302	. . . . . 6 ⊢ ((𝑂:dom 𝑂⟶suc ∪ 𝐴 ∧ Smo 𝑂 ∧ Ord suc ∪ 𝐴) → dom 𝑂 ⊆ suc ∪ 𝐴)
22	13, 16, 20, 21	syl3anc 1374	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ⊆ suc ∪ 𝐴)
23	8, 22	ssexd 5262	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ V)
24		elong 6326	. . . 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 9063	. . . 4 ⊢ (dom 𝑂 ∈ V → dom 𝑂 ≺ 𝒫 dom 𝑂)
28		sdomdom 8921	. . . 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 7723	. . . . . . . . . . 11 ⊢ E We On
32		wess 5611	. . . . . . . . . . 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 5607	. . . . . . . . . 10 ⊢ E Se 𝐴
35	1	oiiso2 9440	. . . . . . . . . 10 ⊢ (( E We 𝐴 ∧ E Se 𝐴) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
36	33, 34, 35	sylancl 587	. . . . . . . . 9 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → 𝑂 Isom E , E (dom 𝑂, ran 𝑂))
37		isof1o 7272	. . . . . . . . 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 6772	. . . . . . . 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 8909	. . . . . . 7 ⊢ ((dom 𝑂 ∈ On ∧ 𝐴 ∈ V ∧ 𝑂:dom 𝑂–1-1-onto→𝐴) → dom 𝑂 ≈ 𝐴)
43	26, 5, 41, 42	syl3anc 1374	. . . . . 6 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≈ 𝐴)
44		endom 8920	. . . . . 6 ⊢ (dom 𝑂 ≈ 𝐴 → dom 𝑂 ≼ 𝐴)
45		domwdom 9483	. . . . . 6 ⊢ (dom 𝑂 ≼ 𝐴 → dom 𝑂 ≼* 𝐴)
46	43, 44, 45	3syl 18	. . . . 5 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐴)
47		wdomtr 9484	. . . . 5 ⊢ ((dom 𝑂 ≼* 𝐴 ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
48	46, 47	sylancom 589	. . . 4 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼* 𝐵)
49		wdompwdom 9487	. . . 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 8948	. . 3 ⊢ ((dom 𝑂 ≼ 𝒫 dom 𝑂 ∧ 𝒫 dom 𝑂 ≼ 𝒫 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
52	29, 50, 51	syl2anc 585	. 2 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ≼ 𝒫 𝐵)
53		elharval 9470	. 2 ⊢ (dom 𝑂 ∈ (har‘𝒫 𝐵) ↔ (dom 𝑂 ∈ On ∧ dom 𝑂 ≼ 𝒫 𝐵))
54	26, 52, 53	sylanbrc 584	1 ⊢ ((𝐴 ⊆ On ∧ 𝐴 ≼* 𝐵) → dom 𝑂 ∈ (har‘𝒫 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  𝒫 cpw 4542  ∪ cuni 4851   class class class wbr 5086   E cep 5524   Se wse 5576   We wwe 5577  dom cdm 5625  ran crn 5626  Ord word 6317  Oncon0 6318  suc csuc 6320  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493   Isom wiso 6494  Smo wsmo 8279   ≈ cen 8884   ≼ cdom 8885   ≺ csdm 8886  OrdIsocoi 9418  harchar 9465   ≼^* cwdom 9473

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-smo 8280  df-recs 8305  df-en 8888  df-dom 8889  df-sdom 8890  df-oi 9419  df-har 9466  df-wdom 9474

This theorem is referenced by:  hsmexlem2  10343  hsmexlem4  10345