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Theorem hsmexlem6 10366
Description: Lemma for hsmex 10367. (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})𝑏𝑋}
hsmexlem4.o 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘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
StepHypRef Expression
1 fvex 6855 . 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})𝑏𝑋}
6 hsmexlem4.o . . . . 5 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
72, 3, 4, 5, 6hsmexlem5 10365 . . . 4 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
85ssrab3 4040 . . . . . 6 𝑆 (𝑅1 “ On)
98sseli 3940 . . . . 5 (𝑑𝑆𝑑 (𝑅1 “ On))
10 harcl 9494 . . . . . 6 (har‘𝒫 (ω × ran 𝐻)) ∈ On
11 r1fnon 9702 . . . . . . 7 𝑅1 Fn On
1211fndmi 6606 . . . . . 6 dom 𝑅1 = On
1310, 12eleqtrri 2837 . . . . 5 (har‘𝒫 (ω × ran 𝐻)) ∈ dom 𝑅1
14 rankr1ag 9737 . . . . 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))))
159, 13, 14sylancl 586 . . . 4 (𝑑𝑆 → (𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ran 𝐻))) ↔ (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻))))
167, 15mpbird 256 . . 3 (𝑑𝑆𝑑 ∈ (𝑅1‘(har‘𝒫 (ω × ran 𝐻))))
1716ssriv 3948 . 2 𝑆 ⊆ (𝑅1‘(har‘𝒫 (ω × ran 𝐻)))
181, 17ssexi 5279 1 𝑆 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1541  wcel 2106  wral 3064  {crab 3407  Vcvv 3445  𝒫 cpw 4560  {csn 4586   cuni 4865   class class class wbr 5105  cmpt 5188   E cep 5536   × cxp 5631  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Oncon0 6317  cfv 6496  ωcom 7801  reccrdg 8354  cdom 8880  OrdIsocoi 9444  harchar 9491  TCctc 9671  𝑅1cr1 9697  rankcrnk 9698
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671  ax-inf2 9576
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7312  df-ov 7359  df-om 7802  df-1st 7920  df-2nd 7921  df-frecs 8211  df-wrecs 8242  df-smo 8291  df-recs 8316  df-rdg 8355  df-en 8883  df-dom 8884  df-sdom 8885  df-oi 9445  df-har 9492  df-wdom 9500  df-tc 9672  df-r1 9699  df-rank 9700
This theorem is referenced by:  hsmex  10367
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