Theorem hsmexlem5 9841

Description: Lemma for hsmex 9843. Combining the above constraints, along with itunitc 9832 and tcrank 9297, 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	1	ssrab3 4008	. . . . . . 7 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On)
3	2	sseli 3911	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On))
4		tcrank 9297	. . . . . 6 ⊢ (𝑑 ∈ ∪ (𝑅1 “ On) → (rank‘𝑑) = (rank “ (TC‘𝑑)))
5	3, 4	syl 17	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = (rank “ (TC‘𝑑)))
6		hsmexlem4.u	. . . . . . . 8 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
7	6	itunitc 9832	. . . . . . 7 ⊢ (TC‘𝑑) = ∪ ran (𝑈‘𝑑)
8	6	itunifn 9828	. . . . . . . 8 ⊢ (𝑑 ∈ 𝑆 → (𝑈‘𝑑) Fn ω)
9		fniunfv 6984	. . . . . . . 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, 10	eqtr4id 2852	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → (TC‘𝑑) = ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐))
12	11	imaeq2d 5896	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank “ (TC‘𝑑)) = (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)))
13		imaiun 6982	. . . . . 6 ⊢ (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))
14	13	a1i 11	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
15	5, 12, 14	3eqtrd 2837	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
16		dmresi 5888	. . . 4 ⊢ dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))
17	15, 16	eqtr4di 2851	. . 3 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
18		rankon 9208	. . . . . 6 ⊢ (rank‘𝑑) ∈ On
19	15, 18	eqeltrrdi 2899	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On)
20		eloni 6169	. . . . 5 ⊢ (∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On → Ord ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
21		oiid 8989	. . . . 5 ⊢ (Ord ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) → OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
22	19, 20, 21	3syl 18	. . . 4 ⊢ (𝑑 ∈ 𝑆 → OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
23	22	dmeqd 5738	. . 3 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
24	17, 23	eqtr4d 2836	. 2 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
25		omex 9090	. . . 4 ⊢ ω ∈ V
26		wdomref 9020	. . . 4 ⊢ (ω ∈ V → ω ≼* ω)
27	25, 26	mp1i 13	. . 3 ⊢ (𝑑 ∈ 𝑆 → ω ≼* ω)
28		frfnom 8053	. . . . . . 7 ⊢ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω
29		hsmexlem4.h	. . . . . . . 8 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
30	29	fneq1i 6420	. . . . . . 7 ⊢ (𝐻 Fn ω ↔ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω)
31	28, 30	mpbir 234	. . . . . 6 ⊢ 𝐻 Fn ω
32		fniunfv 6984	. . . . . 6 ⊢ (𝐻 Fn ω → ∪ 𝑎 ∈ ω (𝐻‘𝑎) = ∪ ran 𝐻)
33	31, 32	ax-mp 5	. . . . 5 ⊢ ∪ 𝑎 ∈ ω (𝐻‘𝑎) = ∪ ran 𝐻
34		iunon 7959	. . . . . . 7 ⊢ ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On) → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
35	25, 34	mpan 689	. . . . . 6 ⊢ (∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
36	29	hsmexlem9 9836	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On)
37	35, 36	mprg 3120	. . . . 5 ⊢ ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On
38	33, 37	eqeltrri 2887	. . . 4 ⊢ ∪ ran 𝐻 ∈ On
39	38	a1i 11	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∪ ran 𝐻 ∈ On)
40		fvssunirn 6674	. . . . . 6 ⊢ (𝐻‘𝑐) ⊆ ∪ ran 𝐻
41		hsmexlem4.x	. . . . . . . 8 ⊢ 𝑋 ∈ V
42		eqid 2798	. . . . . . . 8 ⊢ OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))
43	41, 29, 6, 1, 42	hsmexlem4 9840	. . . . . . 7 ⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ 𝑆) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (𝐻‘𝑐))
44	43	ancoms 462	. . . . . 6 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (𝐻‘𝑐))
45	40, 44	sseldi 3913	. . . . 5 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻)
46		imassrn 5907	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ ran rank
47		rankf 9207	. . . . . . . 8 ⊢ rank:∪ (𝑅1 “ On)⟶On
48		frn 6493	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → ran rank ⊆ On)
49	47, 48	ax-mp 5	. . . . . . 7 ⊢ ran rank ⊆ On
50	46, 49	sstri 3924	. . . . . 6 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ On
51		ffun 6490	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
52		fvex 6658	. . . . . . . . 9 ⊢ ((𝑈‘𝑑)‘𝑐) ∈ V
53	52	funimaex 6411	. . . . . . . 8 ⊢ (Fun rank → (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V)
54	47, 51, 53	mp2b 10	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V
55	54	elpw 4501	. . . . . 6 ⊢ ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ On)
56	50, 55	mpbir 234	. . . . 5 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On
57	45, 56	jctil 523	. . . 4 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻))
58	57	ralrimiva 3149	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻))
59		eqid 2798	. . . 4 ⊢ OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
60	42, 59	hsmexlem3 9839	. . 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)))
61	27, 39, 58, 60	syl21anc 836	. 2 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))
62	24, 61	eqeltrd 2890	1 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  {csn 4525  ∪ cuni 4800  ∪ ciun 4881   class class class wbr 5030   ↦ cmpt 5110   I cid 5424   E cep 5429   × cxp 5517  dom cdm 5519  ran crn 5520   ↾ cres 5521   “ cima 5522  Ord word 6158  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324  ωcom 7560  reccrdg 8028   ≼ cdom 8490  OrdIsocoi 8957  harchar 9004   ≼^* cwdom 9012  TCctc 9162  𝑅₁cr1 9175  rankcrnk 9176

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-se 5479  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-isom 6333  df-riota 7093  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-smo 7966  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-oi 8958  df-har 9005  df-wdom 9013  df-tc 9163  df-r1 9177  df-rank 9178

This theorem is referenced by:  hsmexlem6  9842