Theorem hsmexlem5 10195

Description: Lemma for hsmex 10197. Combining the above constraints, along with itunitc 10186 and tcrank 9651, 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 4016	. . . . . . 7 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On)
3	2	sseli 3918	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On))
4		tcrank 9651	. . . . . 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 10186	. . . . . . 7 ⊢ (TC‘𝑑) = ∪ ran (𝑈‘𝑑)
8	6	itunifn 10182	. . . . . . . 8 ⊢ (𝑑 ∈ 𝑆 → (𝑈‘𝑑) Fn ω)
9		fniunfv 7129	. . . . . . . 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 2798	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → (TC‘𝑑) = ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐))
12	11	imaeq2d 5972	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank “ (TC‘𝑑)) = (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)))
13		imaiun 7127	. . . . . 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 2783	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
16		dmresi 5964	. . . 4 ⊢ dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))
17	15, 16	eqtr4di 2797	. . 3 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
18		rankon 9562	. . . . . 6 ⊢ (rank‘𝑑) ∈ On
19	15, 18	eqeltrrdi 2849	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On)
20		eloni 6280	. . . . 5 ⊢ (∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On → Ord ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
21		oiid 9309	. . . . 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 5817	. . 3 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
24	17, 23	eqtr4d 2782	. 2 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
25		omex 9410	. . . 4 ⊢ ω ∈ V
26		wdomref 9340	. . . 4 ⊢ (ω ∈ V → ω ≼* ω)
27	25, 26	mp1i 13	. . 3 ⊢ (𝑑 ∈ 𝑆 → ω ≼* ω)
28		frfnom 8275	. . . . . . 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 6539	. . . . . . 7 ⊢ (𝐻 Fn ω ↔ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω)
31	28, 30	mpbir 230	. . . . . 6 ⊢ 𝐻 Fn ω
32		fniunfv 7129	. . . . . 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 8179	. . . . . . 7 ⊢ ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On) → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
35	25, 34	mpan 687	. . . . . 6 ⊢ (∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
36	29	hsmexlem9 10190	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On)
37	35, 36	mprg 3079	. . . . 5 ⊢ ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On
38	33, 37	eqeltrri 2837	. . . 4 ⊢ ∪ ran 𝐻 ∈ On
39	38	a1i 11	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∪ ran 𝐻 ∈ On)
40		fvssunirn 6812	. . . . . 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 10194	. . . . . . 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 3920	. . . . 5 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻)
46		imassrn 5983	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ ran rank
47		rankf 9561	. . . . . . . 8 ⊢ rank:∪ (𝑅1 “ On)⟶On
48		frn 6616	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → ran rank ⊆ On)
49	47, 48	ax-mp 5	. . . . . . 7 ⊢ ran rank ⊆ On
50	46, 49	sstri 3931	. . . . . 6 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ On
51		ffun 6612	. . . . . . . 8 ⊢ (rank:∪ (𝑅1 “ On)⟶On → Fun rank)
52		fvex 6796	. . . . . . . . 9 ⊢ ((𝑈‘𝑑)‘𝑐) ∈ V
53	52	funimaex 6529	. . . . . . . 8 ⊢ (Fun rank → (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V)
54	47, 51, 53	mp2b 10	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V
55	54	elpw 4538	. . . . . 6 ⊢ ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ On)
56	50, 55	mpbir 230	. . . . 5 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On
57	45, 56	jctil 520	. . . 4 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻))
58	57	ralrimiva 3104	. . 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 10193	. . 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 835	. 2 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))
62	24, 61	eqeltrd 2840	1 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2107  ∀wral 3065  {crab 3069  Vcvv 3433   ⊆ wss 3888  𝒫 cpw 4534  {csn 4562  ∪ cuni 4840  ∪ ciun 4925   class class class wbr 5075   ↦ cmpt 5158   I cid 5489   E cep 5495   × cxp 5588  dom cdm 5590  ran crn 5591   ↾ cres 5592   “ cima 5593  Ord word 6269  Oncon0 6270  Fun wfun 6431   Fn wfn 6432  ⟶wf 6433  ‘cfv 6437  ωcom 7721  reccrdg 8249   ≼ cdom 8740  OrdIsocoi 9277  harchar 9324   ≼^* cwdom 9332  TCctc 9503  𝑅₁cr1 9529  rankcrnk 9530

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-inf2 9408

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 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-iin 4928  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-smo 8186  df-recs 8211  df-rdg 8250  df-en 8743  df-dom 8744  df-sdom 8745  df-oi 9278  df-har 9325  df-wdom 9333  df-tc 9504  df-r1 9531  df-rank 9532

This theorem is referenced by:  hsmexlem6  10196