Theorem hsmexlem8 10180

Description: Lemma for hsmex 10188. 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 × (𝐻‘𝑎))))

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.

Step	Hyp	Ref	Expression
1		fvex 6787	. 2 ⊢ (har‘𝒫 (𝑋 × (𝐻‘𝑎))) ∈ V
2		hsmexlem7.h	. . 3 ⊢ 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
3		xpeq2 5610	. . . . 5 ⊢ (𝑏 = 𝑧 → (𝑋 × 𝑏) = (𝑋 × 𝑧))
4	3	pweqd 4552	. . . 4 ⊢ (𝑏 = 𝑧 → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × 𝑧))
5	4	fveq2d 6778	. . 3 ⊢ (𝑏 = 𝑧 → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × 𝑧)))
6		xpeq2 5610	. . . . 5 ⊢ (𝑏 = (𝐻‘𝑎) → (𝑋 × 𝑏) = (𝑋 × (𝐻‘𝑎)))
7	6	pweqd 4552	. . . 4 ⊢ (𝑏 = (𝐻‘𝑎) → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × (𝐻‘𝑎)))
8	7	fveq2d 6778	. . 3 ⊢ (𝑏 = (𝐻‘𝑎) → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
9	2, 5, 8	frsucmpt2 8271	. 2 ⊢ ((𝑎 ∈ ω ∧ (har‘𝒫 (𝑋 × (𝐻‘𝑎))) ∈ V) → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
10	1, 9	mpan2 688	1 ⊢ (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  Vcvv 3432  𝒫 cpw 4533   ↦ cmpt 5157   × cxp 5587   ↾ cres 5591  suc csuc 6268  ‘cfv 6433  ωcom 7712  reccrdg 8240  harchar 9315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241

This theorem is referenced by:  hsmexlem9  10181  hsmexlem4  10185