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Theorem hsmexlem5 9135
 Description: Lemma for hsmex 9137. Combining the above constraints, along with itunitc 9126 and tcrank 8630, 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})𝑏𝑋}
hsmexlem4.o 𝑂 = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
Assertion
Ref Expression
hsmexlem5 (𝑑𝑆 → (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
StepHypRef Expression
1 hsmexlem4.s . . . . . . . 8 𝑆 = {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋}
2 ssrab2 3650 . . . . . . . 8 {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋} ⊆ (𝑅1 “ On)
31, 2eqsstri 3598 . . . . . . 7 𝑆 (𝑅1 “ On)
43sseli 3564 . . . . . 6 (𝑑𝑆𝑑 (𝑅1 “ On))
5 tcrank 8630 . . . . . 6 (𝑑 (𝑅1 “ On) → (rank‘𝑑) = (rank “ (TC‘𝑑)))
64, 5syl 17 . . . . 5 (𝑑𝑆 → (rank‘𝑑) = (rank “ (TC‘𝑑)))
7 hsmexlem4.u . . . . . . . . 9 𝑈 = (𝑥 ∈ V ↦ (rec((𝑦 ∈ V ↦ 𝑦), 𝑥) ↾ ω))
87itunifn 9122 . . . . . . . 8 (𝑑𝑆 → (𝑈𝑑) Fn ω)
9 fniunfv 6409 . . . . . . . 8 ((𝑈𝑑) Fn ω → 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
108, 9syl 17 . . . . . . 7 (𝑑𝑆 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
117itunitc 9126 . . . . . . 7 (TC‘𝑑) = ran (𝑈𝑑)
1210, 11syl6reqr 2663 . . . . . 6 (𝑑𝑆 → (TC‘𝑑) = 𝑐 ∈ ω ((𝑈𝑑)‘𝑐))
1312imaeq2d 5385 . . . . 5 (𝑑𝑆 → (rank “ (TC‘𝑑)) = (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)))
14 imaiun 6407 . . . . . 6 (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))
1514a1i 11 . . . . 5 (𝑑𝑆 → (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
166, 13, 153eqtrd 2648 . . . 4 (𝑑𝑆 → (rank‘𝑑) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
17 dmresi 5376 . . . 4 dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))
1816, 17syl6eqr 2662 . . 3 (𝑑𝑆 → (rank‘𝑑) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
19 rankon 8541 . . . . . 6 (rank‘𝑑) ∈ On
2016, 19syl6eqelr 2697 . . . . 5 (𝑑𝑆 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On)
21 eloni 5650 . . . . 5 ( 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On → Ord 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
22 oiid 8329 . . . . 5 (Ord 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) → OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2320, 21, 223syl 18 . . . 4 (𝑑𝑆 → OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2423dmeqd 5248 . . 3 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2518, 24eqtr4d 2647 . 2 (𝑑𝑆 → (rank‘𝑑) = dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
26 omex 8423 . . . 4 ω ∈ V
27 wdomref 8360 . . . 4 (ω ∈ V → ω ≼* ω)
2826, 27mp1i 13 . . 3 (𝑑𝑆 → ω ≼* ω)
29 frfnom 7417 . . . . . . 7 (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω
30 hsmexlem4.h . . . . . . . 8 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
3130fneq1i 5899 . . . . . . 7 (𝐻 Fn ω ↔ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω)
3229, 31mpbir 220 . . . . . 6 𝐻 Fn ω
33 fniunfv 6409 . . . . . 6 (𝐻 Fn ω → 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻)
3432, 33ax-mp 5 . . . . 5 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻
35 iunon 7323 . . . . . . 7 ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻𝑎) ∈ On) → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3626, 35mpan 702 . . . . . 6 (∀𝑎 ∈ ω (𝐻𝑎) ∈ On → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3730hsmexlem9 9130 . . . . . 6 (𝑎 ∈ ω → (𝐻𝑎) ∈ On)
3836, 37mprg 2910 . . . . 5 𝑎 ∈ ω (𝐻𝑎) ∈ On
3934, 38eqeltrri 2685 . . . 4 ran 𝐻 ∈ On
4039a1i 11 . . 3 (𝑑𝑆 ran 𝐻 ∈ On)
41 fvssunirn 6127 . . . . . 6 (𝐻𝑐) ⊆ ran 𝐻
42 hsmexlem4.x . . . . . . . 8 𝑋 ∈ V
43 eqid 2610 . . . . . . . 8 OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
4442, 30, 7, 1, 43hsmexlem4 9134 . . . . . . 7 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ (𝐻𝑐))
4544ancoms 468 . . . . . 6 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ (𝐻𝑐))
4641, 45sseldi 3566 . . . . 5 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻)
47 imassrn 5396 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ ran rank
48 rankf 8540 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
49 frn 5966 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
5048, 49ax-mp 5 . . . . . . 7 ran rank ⊆ On
5147, 50sstri 3577 . . . . . 6 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On
52 ffun 5961 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → Fun rank)
53 fvex 6113 . . . . . . . . 9 ((𝑈𝑑)‘𝑐) ∈ V
5453funimaex 5890 . . . . . . . 8 (Fun rank → (rank “ ((𝑈𝑑)‘𝑐)) ∈ V)
5548, 52, 54mp2b 10 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ∈ V
5655elpw 4114 . . . . . 6 ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On)
5751, 56mpbir 220 . . . . 5 (rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On
5846, 57jctil 558 . . . 4 ((𝑑𝑆𝑐 ∈ ω) → ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
5958ralrimiva 2949 . . 3 (𝑑𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
60 eqid 2610 . . . 4 OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
6143, 60hsmexlem3 9133 . . 3 (((ω ≼* ω ∧ ran 𝐻 ∈ On) ∧ ∀𝑐 ∈ ω ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻)) → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ran 𝐻)))
6228, 40, 59, 61syl21anc 1317 . 2 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ran 𝐻)))
6325, 62eqeltrd 2688 1 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643   E cep 4947   I cid 4948   × cxp 5036  dom cdm 5038  ran crn 5039   ↾ cres 5040   “ cima 5041  Ord word 5639  Oncon0 5640  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  ωcom 6957  reccrdg 7392   ≼ cdom 7839  OrdIsocoi 8297  harchar 8344   ≼* cwdom 8345  TCctc 8495  𝑅1cr1 8508  rankcrnk 8509 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-smo 7330  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-oi 8298  df-har 8346  df-wdom 8347  df-tc 8496  df-r1 8510  df-rank 8511 This theorem is referenced by:  hsmexlem6  9136
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