Theorem hsmexlem5 9252

Description: Lemma for hsmex 9254. Combining the above constraints, along with itunitc 9243 and tcrank 8747, gives an effective constraint on the rank of 𝑆. (Contributed by Stefan O'Rear, 14-Feb-2015.)

Hypotheses

Ref	Expression
hsmexlem4.x	⊢ 𝑋 ∈ V
hsmexlem4.h	⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
hsmexlem4.u	⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
hsmexlem4.s	⊢ 𝑆 = {𝑎 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋}
hsmexlem4.o	⊢ 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))

Assertion

Ref	Expression
hsmexlem5	⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))

Distinct variable groups:   𝑎,𝑐,𝑑,𝐻   𝑆,𝑐,𝑑   𝑈,𝑐,𝑑   𝑎,𝑏,𝑧,𝑋   𝑥,𝑎,𝑦   𝑏,𝑐,𝑑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝑆(𝑥,𝑦,𝑧,𝑎,𝑏)   𝑈(𝑥,𝑦,𝑧,𝑎,𝑏)   𝐻(𝑥,𝑦,𝑧,𝑏)   𝑂(𝑥,𝑦,𝑧,𝑎,𝑏,𝑐,𝑑)   𝑋(𝑥,𝑦,𝑐,𝑑)

Proof of Theorem hsmexlem5

Step	Hyp	Ref	Expression
1		hsmexlem4.s	. . . . . . . 8 ⊢ 𝑆 = {𝑎 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋}
2		ssrab2 3687	. . . . . . . 8 ⊢ {𝑎 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋} ⊆ ∪ (𝑅1 “ On)
3	1, 2	eqsstri 3635	. . . . . . 7 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On)
4	3	sseli 3599	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On))
5		tcrank 8747	. . . . . 6 ⊢ (𝑑 ∈ ∪ (𝑅1 “ On) → (rank‘𝑑) = (rank “ (TC‘𝑑)))
6	4, 5	syl 17	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = (rank “ (TC‘𝑑)))
7		hsmexlem4.u	. . . . . . . . 9 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
8	7	itunifn 9239	. . . . . . . 8 ⊢ (𝑑 ∈ 𝑆 → (𝑈‘𝑑) Fn ω)
9		fniunfv 6505	. . . . . . . 8 ⊢ ((𝑈‘𝑑) Fn ω → ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐) = ∪ ran (𝑈‘𝑑))
10	8, 9	syl 17	. . . . . . 7 ⊢ (𝑑 ∈ 𝑆 → ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐) = ∪ ran (𝑈‘𝑑))
11	7	itunitc 9243	. . . . . . 7 ⊢ (TC‘𝑑) = ∪ ran (𝑈‘𝑑)
12	10, 11	syl6reqr 2675	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → (TC‘𝑑) = ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐))
13	12	imaeq2d 5466	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank “ (TC‘𝑑)) = (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)))
14		imaiun 6503	. . . . . 6 ⊢ (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))
15	14	a1i 11	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
16	6, 13, 15	3eqtrd 2660	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
17		dmresi 5457	. . . 4 ⊢ dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))
18	16, 17	syl6eqr 2674	. . 3 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
19		rankon 8658	. . . . . 6 ⊢ (rank‘𝑑) ∈ On
20	16, 19	syl6eqelr 2710	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On)
21		eloni 5733	. . . . 5 ⊢ (∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On → Ord ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
22		oiid 8446	. . . . 5 ⊢ (Ord ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) → OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
23	20, 21, 22	3syl 18	. . . 4 ⊢ (𝑑 ∈ 𝑆 → OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
24	23	dmeqd 5326	. . 3 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
25	18, 24	eqtr4d 2659	. 2 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
26		omex 8540	. . . 4 ⊢ ω ∈ V
27		wdomref 8477	. . . 4 ⊢ (ω ∈ V → ω ≼* ω)
28	26, 27	mp1i 13	. . 3 ⊢ (𝑑 ∈ 𝑆 → ω ≼* ω)
29		frfnom 7530	. . . . . . 7 ⊢ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω
30		hsmexlem4.h	. . . . . . . 8 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
31	30	fneq1i 5985	. . . . . . 7 ⊢ (𝐻 Fn ω ↔ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω)
32	29, 31	mpbir 221	. . . . . 6 ⊢ 𝐻 Fn ω
33		fniunfv 6505	. . . . . 6 ⊢ (𝐻 Fn ω → ∪ 𝑎 ∈ ω (𝐻‘𝑎) = ∪ ran 𝐻)
34	32, 33	ax-mp 5	. . . . 5 ⊢ ∪ 𝑎 ∈ ω (𝐻‘𝑎) = ∪ ran 𝐻
35		iunon 7436	. . . . . . 7 ⊢ ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On) → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
36	26, 35	mpan 706	. . . . . 6 ⊢ (∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
37	30	hsmexlem9 9247	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On)
38	36, 37	mprg 2926	. . . . 5 ⊢ ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On
39	34, 38	eqeltrri 2698	. . . 4 ⊢ ∪ ran 𝐻 ∈ On
40	39	a1i 11	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∪ ran 𝐻 ∈ On)
41		fvssunirn 6217	. . . . . 6 ⊢ (𝐻‘𝑐) ⊆ ∪ ran 𝐻
42		hsmexlem4.x	. . . . . . . 8 ⊢ 𝑋 ∈ V
43		eqid 2622	. . . . . . . 8 ⊢ OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))
44	42, 30, 7, 1, 43	hsmexlem4 9251	. . . . . . 7 ⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ 𝑆) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (𝐻‘𝑐))
45	44	ancoms 469	. . . . . 6 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (𝐻‘𝑐))
46	41, 45	sseldi 3601	. . . . 5 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻)
47		imassrn 5477	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ ran rank
48		rankf 8657	. . . . . . . 8 ⊢ rank:∪ (𝑅1 “ On)⟶On
49		frn 6053	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → ran rank ⊆ On)
50	48, 49	ax-mp 5	. . . . . . 7 ⊢ ran rank ⊆ On
51	47, 50	sstri 3612	. . . . . 6 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ On
52		ffun 6048	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
53		fvex 6201	. . . . . . . . 9 ⊢ ((𝑈‘𝑑)‘𝑐) ∈ V
54	53	funimaex 5976	. . . . . . . 8 ⊢ (Fun rank → (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V)
55	48, 52, 54	mp2b 10	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V
56	55	elpw 4164	. . . . . 6 ⊢ ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ On)
57	51, 56	mpbir 221	. . . . 5 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On
58	46, 57	jctil 560	. . . 4 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻))
59	58	ralrimiva 2966	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻))
60		eqid 2622	. . . 4 ⊢ OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
61	43, 60	hsmexlem3 9250	. . 3 ⊢ (((ω ≼* ω ∧ ∪ ran 𝐻 ∈ On) ∧ ∀𝑐 ∈ ω ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻)) → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))
62	28, 40, 59, 61	syl21anc 1325	. 2 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))
63	25, 62	eqeltrd 2701	1 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   I cid 5023   E cep 5028   × cxp 5112  dom cdm 5114  ran crn 5115   ↾ cres 5116   “ cima 5117  Ord word 5722  Oncon0 5723  Fun wfun 5882   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  ωcom 7065  reccrdg 7505   ≼ cdom 7953  OrdIsocoi 8414  harchar 8461   ≼^* cwdom 8462  TCctc 8612  𝑅₁cr1 8625  rankcrnk 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-oi 8415  df-har 8463  df-wdom 8464  df-tc 8613  df-r1 8627  df-rank 8628

This theorem is referenced by:  hsmexlem6  9253