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Theorem hsmexlem8 10462
Description: Lemma for hsmex 10470. 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 6920 . 2 (har‘𝒫 (𝑋 × (𝐻𝑎))) ∈ V
2 hsmexlem7.h . . 3 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
3 xpeq2 5710 . . . . 5 (𝑏 = 𝑧 → (𝑋 × 𝑏) = (𝑋 × 𝑧))
43pweqd 4622 . . . 4 (𝑏 = 𝑧 → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × 𝑧))
54fveq2d 6911 . . 3 (𝑏 = 𝑧 → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × 𝑧)))
6 xpeq2 5710 . . . . 5 (𝑏 = (𝐻𝑎) → (𝑋 × 𝑏) = (𝑋 × (𝐻𝑎)))
76pweqd 4622 . . . 4 (𝑏 = (𝐻𝑎) → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × (𝐻𝑎)))
87fveq2d 6911 . . 3 (𝑏 = (𝐻𝑎) → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
92, 5, 8frsucmpt2 8479 . 2 ((𝑎 ∈ ω ∧ (har‘𝒫 (𝑋 × (𝐻𝑎))) ∈ V) → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
101, 9mpan2 691 1 (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻𝑎))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  𝒫 cpw 4605  cmpt 5231   × cxp 5687  cres 5691  suc csuc 6388  cfv 6563  ωcom 7887  reccrdg 8448  harchar 9594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449
This theorem is referenced by:  hsmexlem9  10463  hsmexlem4  10467
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