Theorem hsmexlem8 10335

Description: Lemma for hsmex 10343. 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 6842	. 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 5641	. . . . 5 ⊢ (𝑏 = 𝑧 → (𝑋 × 𝑏) = (𝑋 × 𝑧))
4	3	pweqd 4548	. . . 4 ⊢ (𝑏 = 𝑧 → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × 𝑧))
5	4	fveq2d 6833	. . 3 ⊢ (𝑏 = 𝑧 → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × 𝑧)))
6		xpeq2 5641	. . . . 5 ⊢ (𝑏 = (𝐻‘𝑎) → (𝑋 × 𝑏) = (𝑋 × (𝐻‘𝑎)))
7	6	pweqd 4548	. . . 4 ⊢ (𝑏 = (𝐻‘𝑎) → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × (𝐻‘𝑎)))
8	7	fveq2d 6833	. . 3 ⊢ (𝑏 = (𝐻‘𝑎) → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
9	2, 5, 8	frsucmpt2 8368	. 2 ⊢ ((𝑎 ∈ ω ∧ (har‘𝒫 (𝑋 × (𝐻‘𝑎))) ∈ V) → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
10	1, 9	mpan2 692	1 ⊢ (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3427  𝒫 cpw 4531   ↦ cmpt 5155   × cxp 5618   ↾ cres 5622  suc csuc 6314  ‘cfv 6487  ωcom 7806  reccrdg 8337  harchar 9460

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6254  df-ord 6315  df-on 6316  df-lim 6317  df-suc 6318  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-om 7807  df-2nd 7932  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338

This theorem is referenced by:  hsmexlem9  10336  hsmexlem4  10340