Theorem hsmexlem8 10342

Description: Lemma for hsmex 10350. 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 6843	. 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 6834	. . 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 6834	. . 3 ⊢ (𝑏 = (𝐻‘𝑎) → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
9	2, 5, 8	frsucmpt2 8373	. 2 ⊢ ((𝑎 ∈ ω ∧ (har‘𝒫 (𝑋 × (𝐻‘𝑎))) ∈ V) → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
10	1, 9	mpan2 698	1 ⊢ (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))

Syntax hints:   → wi 4   = wceq 1548   ∈ wcel 2121  Vcvv 3433  𝒫 cpw 4531   ↦ cmpt 5155   × cxp 5618   ↾ cres 5622  suc csuc 6315  ‘cfv 6488  ωcom 7809  reccrdg 8342  harchar 9465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  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 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-om 7810  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343

This theorem is referenced by:  hsmexlem9  10343  hsmexlem4  10347