Theorem hsmexlem6 10347

Description: Lemma for hsmex 10348. (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 6848	. 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 10346	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻)))
8	5	ssrab3 4023	. . . . . 6 ⊢ 𝑆 ⊆ ∪ (𝑅1 “ On)
9	8	sseli 3918	. . . . 5 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ ∪ (𝑅1 “ On))
10		harcl 9468	. . . . . 6 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ On
11		r1fnon 9685	. . . . . . 7 ⊢ 𝑅1 Fn On
12	11	fndmi 6597	. . . . . 6 ⊢ dom 𝑅1 = On
13	10, 12	eleqtrri 2836	. . . . 5 ⊢ (har‘𝒫 (ω × ∪ ran 𝐻)) ∈ dom 𝑅1
14		rankr1ag 9720	. . . . 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))))
15	9, 13, 14	sylancl 587	. . . 4 ⊢ (𝑑 ∈ 𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ∪ ran 𝐻))))
16	7, 15	mpbird 257	. . 3 ⊢ (𝑑 ∈ 𝑆 → 𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻))))
17	16	ssriv 3926	. 2 ⊢ 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ∪ ran 𝐻)))
18	1, 17	ssexi 5260	1 ⊢ 𝑆 ∈ V

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  ∀wral 3052  {crab 3390  Vcvv 3430  𝒫 cpw 4542  {csn 4568  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167   E cep 5524   × cxp 5623  dom cdm 5625  ran crn 5626   ↾ cres 5627   “ cima 5628  Oncon0 6318  ‘cfv 6493  ωcom 7811  reccrdg 8342   ≼ cdom 8885  OrdIsocoi 9418  harchar 9465  TCctc 9649  𝑅₁cr1 9680  rankcrnk 9681

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-smo 8280  df-recs 8305  df-rdg 8343  df-en 8888  df-dom 8889  df-sdom 8890  df-oi 9419  df-har 9466  df-wdom 9474  df-tc 9650  df-r1 9682  df-rank 9683

This theorem is referenced by:  hsmex  10348