Theorem hsmexlem5 10343

Description: Lemma for hsmex 10345. Combining the above constraints, along with itunitc 10334 and tcrank 9799, 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 4013	. . . . . . 7 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On)
3	2	sseli 3911	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On))
4		tcrank 9799	. . . . . 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 10334	. . . . . . 7 ⊢ (TC‘𝑑) = ∪ ran (𝑈‘𝑑)
8	6	itunifn 10330	. . . . . . . 8 ⊢ (𝑑 ∈ 𝑆 → (𝑈‘𝑑) Fn ω)
9		fniunfv 7191	. . . . . . . 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 2793	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → (TC‘𝑑) = ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐))
12	11	imaeq2d 6012	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank “ (TC‘𝑑)) = (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)))
13		imaiun 7189	. . . . . 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 2778	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
16		dmresi 6004	. . . 4 ⊢ dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))
17	15, 16	eqtr4di 2792	. . 3 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
18		rankon 9710	. . . . . 6 ⊢ (rank‘𝑑) ∈ On
19	15, 18	eqeltrrdi 2848	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On)
20		eloni 6320	. . . . 5 ⊢ (∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On → Ord ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
21		oiid 9446	. . . . 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 5847	. . 3 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
24	17, 23	eqtr4d 2777	. 2 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
25		omex 9555	. . . 4 ⊢ ω ∈ V
26		wdomref 9477	. . . 4 ⊢ (ω ∈ V → ω ≼* ω)
27	25, 26	mp1i 13	. . 3 ⊢ (𝑑 ∈ 𝑆 → ω ≼* ω)
28		frfnom 8364	. . . . . . 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 6582	. . . . . . 7 ⊢ (𝐻 Fn ω ↔ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω)
31	28, 30	mpbir 232	. . . . . 6 ⊢ 𝐻 Fn ω
32		fniunfv 7191	. . . . . 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 8269	. . . . . . 7 ⊢ ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On) → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
35	25, 34	mpan 696	. . . . . 6 ⊢ (∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
36	29	hsmexlem9 10338	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On)
37	35, 36	mprg 3059	. . . . 5 ⊢ ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On
38	33, 37	eqeltrri 2836	. . . 4 ⊢ ∪ ran 𝐻 ∈ On
39	38	a1i 11	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∪ ran 𝐻 ∈ On)
40		fvssunirn 6858	. . . . . 6 ⊢ (𝐻‘𝑐) ⊆ ∪ ran 𝐻
41		hsmexlem4.x	. . . . . . . 8 ⊢ 𝑋 ∈ V
42		eqid 2739	. . . . . . . 8 ⊢ OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))
43	41, 29, 6, 1, 42	hsmexlem4 10342	. . . . . . 7 ⊢ ((𝑐 ∈ ω ∧ 𝑑 ∈ 𝑆) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (𝐻‘𝑐))
44	43	ancoms 459	. . . . . 6 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (𝐻‘𝑐))
45	40, 44	sselid 3913	. . . . 5 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻)
46		imassrn 6023	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ ran rank
47		rankf 9709	. . . . . . . 8 ⊢ rank:∪ (𝑅1 “ On)⟶On
48		frn 6662	. . . . . . . 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 6658	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
52		fvex 6840	. . . . . . . . 9 ⊢ ((𝑈‘𝑑)‘𝑐) ∈ V
53	52	funimaex 6573	. . . . . . . 8 ⊢ (Fun rank → (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V)
54	47, 51, 53	mp2b 10	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V
55	54	elpw 4533	. . . . . 6 ⊢ ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ On)
56	50, 55	mpbir 232	. . . . 5 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On
57	45, 56	jctil 524	. . . 4 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻))
58	57	ralrimiva 3131	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻))
59		eqid 2739	. . . 4 ⊢ OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
60	42, 59	hsmexlem3 10341	. . 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 843	. 2 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))
62	24, 61	eqeltrd 2839	1 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3053  {crab 3391  Vcvv 3431   ⊆ wss 3883  𝒫 cpw 4529  {csn 4555  ∪ cuni 4838  ∪ ciun 4921   class class class wbr 5072   ↦ cmpt 5153   I cid 5512   E cep 5517   × cxp 5616  dom cdm 5618  ran crn 5619   ↾ cres 5620   “ cima 5621  Ord word 6309  Oncon0 6310  Fun wfun 6479   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  ωcom 7806  reccrdg 8338   ≼ cdom 8881  OrdIsocoi 9414  harchar 9461   ≼^* cwdom 9469  TCctc 9646  𝑅₁cr1 9677  rankcrnk 9678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-smo 8276  df-recs 8301  df-rdg 8339  df-en 8884  df-dom 8885  df-sdom 8886  df-oi 9415  df-har 9462  df-wdom 9470  df-tc 9647  df-r1 9679  df-rank 9680

This theorem is referenced by:  hsmexlem6  10344