Theorem hsmexlem8 10429

Description: Lemma for hsmex 10437. 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 6895	. 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 5680	. . . . 5 ⊢ (𝑏 = 𝑧 → (𝑋 × 𝑏) = (𝑋 × 𝑧))
4	3	pweqd 4577	. . . 4 ⊢ (𝑏 = 𝑧 → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × 𝑧))
5	4	fveq2d 6886	. . 3 ⊢ (𝑏 = 𝑧 → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × 𝑧)))
6		xpeq2 5680	. . . . 5 ⊢ (𝑏 = (𝐻‘𝑎) → (𝑋 × 𝑏) = (𝑋 × (𝐻‘𝑎)))
7	6	pweqd 4577	. . . 4 ⊢ (𝑏 = (𝐻‘𝑎) → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × (𝐻‘𝑎)))
8	7	fveq2d 6886	. . 3 ⊢ (𝑏 = (𝐻‘𝑎) → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
9	2, 5, 8	frsucmpt2 8432	. 2 ⊢ ((𝑎 ∈ ω ∧ (har‘𝒫 (𝑋 × (𝐻‘𝑎))) ∈ V) → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
10	1, 9	mpan2 704	1 ⊢ (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453  𝒫 cpw 4560   ↦ cmpt 5190   × cxp 5657   ↾ cres 5661  suc csuc 6363  ‘cfv 6537  ωcom 7865  reccrdg 8401  harchar 9531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402

This theorem is used by:  hsmexlem9  10430  hsmexlem4  10434