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Theorem hsmexlem5 10427

Description: Lemma for hsmex 10429. Combining the above constraints, along with itunitc 10418 and tcrank 9881, 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 4080	. . . . . . 7 ⊢ 𝑆 ⊆ ∪ (𝑅₁ “ On)
3	2	sseli 3978	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅₁ “ On))
4		tcrank 9881	. . . . . 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 10418	. . . . . . 7 ⊢ (TC‘𝑑) = ∪ ran (𝑈‘𝑑)
8	6	itunifn 10414	. . . . . . . 8 ⊢ (𝑑 ∈ 𝑆 → (𝑈‘𝑑) Fn ω)
9		fniunfv 7248	. . . . . . . 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 2791	. . . . . 6 ⊢ (𝑑 ∈ 𝑆 → (TC‘𝑑) = ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐))
12	11	imaeq2d 6059	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → (rank “ (TC‘𝑑)) = (rank “ ∪ 𝑐 ∈ ω ((𝑈‘𝑑)‘𝑐)))
13		imaiun 7246	. . . . . 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 2776	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
16		dmresi 6051	. . . 4 ⊢ dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))
17	15, 16	eqtr4di 2790	. . 3 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
18		rankon 9792	. . . . . 6 ⊢ (rank‘𝑑) ∈ On
19	15, 18	eqeltrrdi 2842	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On)
20		eloni 6374	. . . . 5 ⊢ (∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ On → Ord ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
21		oiid 9538	. . . . 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 5905	. . 3 ⊢ (𝑑 ∈ 𝑆 → dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = dom ( I ↾ ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
24	17, 23	eqtr4d 2775	. 2 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) = dom OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))))
25		omex 9640	. . . 4 ⊢ ω ∈ V
26		wdomref 9569	. . . 4 ⊢ (ω ∈ V → ω ≼^* ω)
27	25, 26	mp1i 13	. . 3 ⊢ (𝑑 ∈ 𝑆 → ω ≼^* ω)
28		frfnom 8437	. . . . . . 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 6646	. . . . . . 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 7248	. . . . . 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 8341	. . . . . . 7 ⊢ ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On) → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
35	25, 34	mpan 688	. . . . . 6 ⊢ (∀𝑎 ∈ ω (𝐻‘𝑎) ∈ On → ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On)
36	29	hsmexlem9 10422	. . . . . 6 ⊢ (𝑎 ∈ ω → (𝐻‘𝑎) ∈ On)
37	35, 36	mprg 3067	. . . . 5 ⊢ ∪ 𝑎 ∈ ω (𝐻‘𝑎) ∈ On
38	33, 37	eqeltrri 2830	. . . 4 ⊢ ∪ ran 𝐻 ∈ On
39	38	a1i 11	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∪ ran 𝐻 ∈ On)
40		fvssunirn 6924	. . . . . 6 ⊢ (𝐻‘𝑐) ⊆ ∪ ran 𝐻
41		hsmexlem4.x	. . . . . . . 8 ⊢ 𝑋 ∈ V
42		eqid 2732	. . . . . . . 8 ⊢ OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))
43	41, 29, 6, 1, 42	hsmexlem4 10426	. . . . . . 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 3980	. . . . 5 ⊢ ((𝑑 ∈ 𝑆 ∧ 𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻)
46		imassrn 6070	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ ran rank
47		rankf 9791	. . . . . . . 8 ⊢ rank:∪ (𝑅₁ “ On)⟶On
48		frn 6724	. . . . . . . 8 ⊢ (rank:∪ (𝑅₁ “ On)⟶On → ran rank ⊆ On)
49	47, 48	ax-mp 5	. . . . . . 7 ⊢ ran rank ⊆ On
50	46, 49	sstri 3991	. . . . . 6 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ⊆ On
51		ffun 6720	. . . . . . . 8 ⊢ (rank:∪ (𝑅₁ “ On)⟶On → Fun rank)
52		fvex 6904	. . . . . . . . 9 ⊢ ((𝑈‘𝑑)‘𝑐) ∈ V
53	52	funimaex 6636	. . . . . . . 8 ⊢ (Fun rank → (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V)
54	47, 51, 53	mp2b 10	. . . . . . 7 ⊢ (rank “ ((𝑈‘𝑑)‘𝑐)) ∈ V
55	54	elpw 4606	. . . . . 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 3146	. . 3 ⊢ (𝑑 ∈ 𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈‘𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐))) ∈ ∪ ran 𝐻))
59		eqid 2732	. . . 4 ⊢ OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐))) = OrdIso( E , ∪ 𝑐 ∈ ω (rank “ ((𝑈‘𝑑)‘𝑐)))
60	42, 59	hsmexlem3 10425	. . 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 2833	1 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 Vcvv 3474 ⊆ wss 3948 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 E cep 5579 × cxp 5674 dom cdm 5676 ran crn 5677 ↾ cres 5678 “ cima 5679 Ord word 6363 Oncon0 6364 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 ‘cfv 6543 ωcom 7857 reccrdg 8411 ≼ cdom 8939 OrdIsocoi 9506 harchar 9553 ≼^* cwdom 9561 TCctc 9733 𝑅₁cr1 9759 rankcrnk 9760

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-smo 8348 df-recs 8373 df-rdg 8412 df-en 8942 df-dom 8943 df-sdom 8944 df-oi 9507 df-har 9554 df-wdom 9562 df-tc 9734 df-r1 9761 df-rank 9762

This theorem is referenced by: hsmexlem6 10428

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