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Theorem hsmexlem8 9835
Description: Lemma for hsmex 9843. Properties of the recurrent sequence of ordinals. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Hypothesis
Ref Expression
hsmexlem7.h 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
Assertion
Ref Expression
hsmexlem8 (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
Distinct variable groups:   𝑧,𝑋   𝑧,𝑎
Allowed substitution hints:   𝐻(𝑧,𝑎)   𝑋(𝑎)

Proof of Theorem hsmexlem8
Dummy variable 𝑏 is distinct from all other variables.
StepHypRef Expression
1 fvex 6665 . 2 (har‘𝒫 (𝑋 × (𝐻𝑎))) ∈ V
2 hsmexlem7.h . . 3 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
3 xpeq2 5553 . . . . 5 (𝑏 = 𝑧 → (𝑋 × 𝑏) = (𝑋 × 𝑧))
43pweqd 4530 . . . 4 (𝑏 = 𝑧 → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × 𝑧))
54fveq2d 6656 . . 3 (𝑏 = 𝑧 → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × 𝑧)))
6 xpeq2 5553 . . . . 5 (𝑏 = (𝐻𝑎) → (𝑋 × 𝑏) = (𝑋 × (𝐻𝑎)))
76pweqd 4530 . . . 4 (𝑏 = (𝐻𝑎) → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × (𝐻𝑎)))
87fveq2d 6656 . . 3 (𝑏 = (𝐻𝑎) → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
92, 5, 8frsucmpt2 8063 . 2 ((𝑎 ∈ ω ∧ (har‘𝒫 (𝑋 × (𝐻𝑎))) ∈ V) → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
101, 9mpan2 690 1 (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2114  Vcvv 3469  𝒫 cpw 4511  cmpt 5122   × cxp 5530  cres 5534  suc csuc 6171  cfv 6334  ωcom 7565  reccrdg 8032  harchar 9008
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-om 7566  df-wrecs 7934  df-recs 7995  df-rdg 8033
This theorem is referenced by:  hsmexlem9  9836  hsmexlem4  9840
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