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Theorem hsmexlem5 9112
Description: Lemma for hsmex 9114. Combining the above constraints, along with itunitc 9103 and tcrank 8607, 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 3649 . . . . . . . 8 {𝑎 (𝑅1 “ On) ∣ ∀𝑏 ∈ (TC‘{𝑎})𝑏𝑋} ⊆ (𝑅1 “ On)
31, 2eqsstri 3597 . . . . . . 7 𝑆 (𝑅1 “ On)
43sseli 3563 . . . . . 6 (𝑑𝑆𝑑 (𝑅1 “ On))
5 tcrank 8607 . . . . . 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 9099 . . . . . . . 8 (𝑑𝑆 → (𝑈𝑑) Fn ω)
9 fniunfv 6387 . . . . . . . 8 ((𝑈𝑑) Fn ω → 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
108, 9syl 17 . . . . . . 7 (𝑑𝑆 𝑐 ∈ ω ((𝑈𝑑)‘𝑐) = ran (𝑈𝑑))
117itunitc 9103 . . . . . . 7 (TC‘𝑑) = ran (𝑈𝑑)
1210, 11syl6reqr 2662 . . . . . 6 (𝑑𝑆 → (TC‘𝑑) = 𝑐 ∈ ω ((𝑈𝑑)‘𝑐))
1312imaeq2d 5372 . . . . 5 (𝑑𝑆 → (rank “ (TC‘𝑑)) = (rank “ 𝑐 ∈ ω ((𝑈𝑑)‘𝑐)))
14 imaiun 6385 . . . . . 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 2647 . . . 4 (𝑑𝑆 → (rank‘𝑑) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
17 dmresi 5363 . . . 4 dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))
1816, 17syl6eqr 2661 . . 3 (𝑑𝑆 → (rank‘𝑑) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
19 rankon 8518 . . . . . 6 (rank‘𝑑) ∈ On
2016, 19syl6eqelr 2696 . . . . 5 (𝑑𝑆 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On)
21 eloni 5636 . . . . 5 ( 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)) ∈ On → Ord 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
22 oiid 8306 . . . . 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 5235 . . 3 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = dom ( I ↾ 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
2518, 24eqtr4d 2646 . 2 (𝑑𝑆 → (rank‘𝑑) = dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))))
26 omex 8400 . . . 4 ω ∈ V
27 wdomref 8337 . . . 4 (ω ∈ V → ω ≼* ω)
2826, 27mp1i 13 . . 3 (𝑑𝑆 → ω ≼* ω)
29 frfnom 7394 . . . . . . 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 5885 . . . . . . 7 (𝐻 Fn ω ↔ (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω) Fn ω)
3229, 31mpbir 219 . . . . . 6 𝐻 Fn ω
33 fniunfv 6387 . . . . . 6 (𝐻 Fn ω → 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻)
3432, 33ax-mp 5 . . . . 5 𝑎 ∈ ω (𝐻𝑎) = ran 𝐻
35 iunon 7300 . . . . . . 7 ((ω ∈ V ∧ ∀𝑎 ∈ ω (𝐻𝑎) ∈ On) → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3626, 35mpan 701 . . . . . 6 (∀𝑎 ∈ ω (𝐻𝑎) ∈ On → 𝑎 ∈ ω (𝐻𝑎) ∈ On)
3730hsmexlem9 9107 . . . . . 6 (𝑎 ∈ ω → (𝐻𝑎) ∈ On)
3836, 37mprg 2909 . . . . 5 𝑎 ∈ ω (𝐻𝑎) ∈ On
3934, 38eqeltrri 2684 . . . 4 ran 𝐻 ∈ On
4039a1i 11 . . 3 (𝑑𝑆 ran 𝐻 ∈ On)
41 fvssunirn 6112 . . . . . 6 (𝐻𝑐) ⊆ ran 𝐻
42 hsmexlem4.x . . . . . . . 8 𝑋 ∈ V
43 eqid 2609 . . . . . . . 8 OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐)))
4442, 30, 7, 1, 43hsmexlem4 9111 . . . . . . 7 ((𝑐 ∈ ω ∧ 𝑑𝑆) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ (𝐻𝑐))
4544ancoms 467 . . . . . 6 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ (𝐻𝑐))
4641, 45sseldi 3565 . . . . 5 ((𝑑𝑆𝑐 ∈ ω) → dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻)
47 imassrn 5383 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ ran rank
48 rankf 8517 . . . . . . . 8 rank: (𝑅1 “ On)⟶On
49 frn 5952 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → ran rank ⊆ On)
5048, 49ax-mp 5 . . . . . . 7 ran rank ⊆ On
5147, 50sstri 3576 . . . . . 6 (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On
52 ffun 5947 . . . . . . . 8 (rank: (𝑅1 “ On)⟶On → Fun rank)
53 fvex 6098 . . . . . . . . 9 ((𝑈𝑑)‘𝑐) ∈ V
5453funimaex 5876 . . . . . . . 8 (Fun rank → (rank “ ((𝑈𝑑)‘𝑐)) ∈ V)
5548, 52, 54mp2b 10 . . . . . . 7 (rank “ ((𝑈𝑑)‘𝑐)) ∈ V
5655elpw 4113 . . . . . 6 ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ↔ (rank “ ((𝑈𝑑)‘𝑐)) ⊆ On)
5751, 56mpbir 219 . . . . 5 (rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On
5846, 57jctil 557 . . . 4 ((𝑑𝑆𝑐 ∈ ω) → ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
5958ralrimiva 2948 . . 3 (𝑑𝑆 → ∀𝑐 ∈ ω ((rank “ ((𝑈𝑑)‘𝑐)) ∈ 𝒫 On ∧ dom OrdIso( E , (rank “ ((𝑈𝑑)‘𝑐))) ∈ ran 𝐻))
60 eqid 2609 . . . 4 OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) = OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐)))
6143, 60hsmexlem3 9110 . . 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 1316 . 2 (𝑑𝑆 → dom OrdIso( E , 𝑐 ∈ ω (rank “ ((𝑈𝑑)‘𝑐))) ∈ (har‘𝒫 (ω × ran 𝐻)))
6325, 62eqeltrd 2687 1 (𝑑𝑆 → (rank‘𝑑) ∈ (har‘𝒫 (ω × ran 𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  wss 3539  𝒫 cpw 4107  {csn 4124   cuni 4366   ciun 4449   class class class wbr 4577  cmpt 4637   E cep 4937   I cid 4938   × cxp 5026  dom cdm 5028  ran crn 5029  cres 5030  cima 5031  Ord word 5625  Oncon0 5626  Fun wfun 5784   Fn wfn 5785  wf 5786  cfv 5790  ωcom 6934  reccrdg 7369  cdom 7816  OrdIsocoi 8274  harchar 8321  * cwdom 8322  TCctc 8472  𝑅1cr1 8485  rankcrnk 8486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-inf2 8398
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-se 4988  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-isom 5799  df-riota 6489  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-smo 7307  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-oi 8275  df-har 8323  df-wdom 8324  df-tc 8473  df-r1 8487  df-rank 8488
This theorem is referenced by:  hsmexlem6  9113
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