Theorem hsmexlem6 9847

Description: Lemma for hsmex 9848. (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
hsmexlem6	⊢ 𝑆 ∈ V

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 hsmexlem6

Step	Hyp	Ref	Expression
1		fvex 6677	. 2 ⊢ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ∈ V
2		hsmexlem4.x	. . . . 5 ⊢ 𝑋 ∈ V
3		hsmexlem4.h	. . . . 5 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
4		hsmexlem4.u	. . . . 5 ⊢ 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ ∪ 𝑦), 𝑥) ↾ ω))
5		hsmexlem4.s	. . . . 5 ⊢ 𝑆 = {𝑎 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏 ≼ 𝑋}
6		hsmexlem4.o	. . . . 5 ⊢ 𝑂 = OrdIso( E , (rank “ ((𝑈‘𝑑)‘𝑐)))
7	2, 3, 4, 5, 6	hsmexlem5 9846	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))
8	5	ssrab3 4056	. . . . . 6 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On)
9	8	sseli 3962	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On))
10		harcl 9019	. . . . . 6 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ On
11		r1fnon 9190	. . . . . . 7 ⊢ 𝑅1 Fn On
12		fndm 6449	. . . . . . 7 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
13	11, 12	ax-mp 5	. . . . . 6 ⊢ dom 𝑅1 = On
14	10, 13	eleqtrri 2912	. . . . 5 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1
15		rankr1ag 9225	. . . . 5 ⊢ ((𝑑 ∈ ∪ (𝑅1 “ On) ∧ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1) → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻))))
16	9, 14, 15	sylancl 588	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻))))
17	7, 16	mpbird 259	. . 3 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))))
18	17	ssriv 3970	. 2 ⊢ 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻)))
19	1, 18	ssexi 5218	1 ⊢ 𝑆 ∈ V

Syntax hints:   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142  Vcvv 3494  𝒫 cpw 4538  {csn 4560  ∪ cuni 4831   class class class wbr 5058   ↦ cmpt 5138   E cep 5458   × cxp 5547  dom cdm 5549  ran crn 5550   ↾ cres 5551   “ cima 5552  Oncon0 6185   Fn wfn 6344  ‘cfv 6349  ωcom 7574  reccrdg 8039   ≼ cdom 8501  OrdIsocoi 8967  harchar 9014  TCctc 9172  𝑅₁cr1 9185  rankcrnk 9186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-inf2 9098

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-iin 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-smo 7977  df-recs 8002  df-rdg 8040  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-oi 8968  df-har 9016  df-wdom 9017  df-tc 9173  df-r1 9187  df-rank 9188

This theorem is referenced by:  hsmex  9848