Theorem hsmexlem8 10111

Description: Lemma for hsmex 10119. 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 6769	. 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 5601	. . . . 5 ⊢ (𝑏 = 𝑧 → (𝑋 × 𝑏) = (𝑋 × 𝑧))
4	3	pweqd 4549	. . . 4 ⊢ (𝑏 = 𝑧 → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × 𝑧))
5	4	fveq2d 6760	. . 3 ⊢ (𝑏 = 𝑧 → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × 𝑧)))
6		xpeq2 5601	. . . . 5 ⊢ (𝑏 = (𝐻‘𝑎) → (𝑋 × 𝑏) = (𝑋 × (𝐻‘𝑎)))
7	6	pweqd 4549	. . . 4 ⊢ (𝑏 = (𝐻‘𝑎) → 𝒫 (𝑋 × 𝑏) = 𝒫 (𝑋 × (𝐻‘𝑎)))
8	7	fveq2d 6760	. . 3 ⊢ (𝑏 = (𝐻‘𝑎) → (har‘𝒫 (𝑋 × 𝑏)) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
9	2, 5, 8	frsucmpt2 8241	. 2 ⊢ ((𝑎 ∈ ω ∧ (har‘𝒫 (𝑋 × (𝐻‘𝑎))) ∈ V) → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))
10	1, 9	mpan2 687	1 ⊢ (𝑎 ∈ ω → (𝐻‘suc 𝑎) = (har‘𝒫 (𝑋 × (𝐻‘𝑎))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  𝒫 cpw 4530   ↦ cmpt 5153   × cxp 5578   ↾ cres 5582  suc csuc 6253  ‘cfv 6418  ωcom 7687  reccrdg 8211  harchar 9245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212

This theorem is referenced by:  hsmexlem9  10112  hsmexlem4  10116