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Theorem hsmexlem7 9834
 Description: Lemma for hsmex 9843. 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
hsmexlem7 (𝐻‘∅) = (har‘𝒫 𝑋)
Distinct variable group:   𝑧,𝑋
Allowed substitution hint:   𝐻(𝑧)

Proof of Theorem hsmexlem7
StepHypRef Expression
1 hsmexlem7.h . . 3 𝐻 = (rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)
21fveq1i 6668 . 2 (𝐻‘∅) = ((rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)‘∅)
3 fvex 6680 . . 3 (har‘𝒫 𝑋) ∈ V
4 fr0g 8062 . . 3 ((har‘𝒫 𝑋) ∈ V → ((rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)‘∅) = (har‘𝒫 𝑋))
53, 4ax-mp 5 . 2 ((rec((𝑧 ∈ V ↦ (har‘𝒫 (𝑋 × 𝑧))), (har‘𝒫 𝑋)) ↾ ω)‘∅) = (har‘𝒫 𝑋)
62, 5eqtri 2849 1 (𝐻‘∅) = (har‘𝒫 𝑋)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1530   ∈ wcel 2107  Vcvv 3500  ∅c0 4295  𝒫 cpw 4542   ↦ cmpt 5143   × cxp 5552   ↾ cres 5556  ‘cfv 6352  ωcom 7568  reccrdg 8036  harchar 9009 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-wrecs 7938  df-recs 7999  df-rdg 8037 This theorem is referenced by:  hsmexlem9  9836  hsmexlem4  9840
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