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Theorem hsmexlem8 9458
 Description: Lemma for hsmex 9466. 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 6363 . 2 (har‘𝒫 (𝑋 × (𝐻𝑎))) ∈ V
2 hsmexlem7.h . . 3 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
3 xpeq2 5286 . . . . 5 (𝑏 = 𝑧 → (𝑋 × 𝑏) = (𝑋 × 𝑧))
43pweqd 4307 . . . 4 (𝑏 = 𝑧 → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × 𝑧))
54fveq2d 6357 . . 3 (𝑏 = 𝑧 → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × 𝑧)))
6 xpeq2 5286 . . . . 5 (𝑏 = (𝐻𝑎) → (𝑋 × 𝑏) = (𝑋 × (𝐻𝑎)))
76pweqd 4307 . . . 4 (𝑏 = (𝐻𝑎) → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × (𝐻𝑎)))
87fveq2d 6357 . . 3 (𝑏 = (𝐻𝑎) → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
92, 5, 8frsucmpt2 7705 . 2 ((𝑎 ∈ ω ∧ (har‘𝒫 (𝑋 × (𝐻𝑎))) ∈ V) → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
101, 9mpan2 709 1 (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340  𝒫 cpw 4302   ↦ cmpt 4881   × cxp 5264   ↾ cres 5268  suc csuc 5886  ‘cfv 6049  ωcom 7231  reccrdg 7675  harchar 8628 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-wrecs 7577  df-recs 7638  df-rdg 7676 This theorem is referenced by:  hsmexlem9  9459  hsmexlem4  9463
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