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Theorem hsmexlem3 9288
 Description: Lemma for hsmex 9292. Clear 𝐼 hypothesis and extend previous result by dominance. Note that this could be substantially strengthened, e.g. using the weak Hartogs function, but all we need here is that there be *some* dominating ordinal. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)
Hypotheses
Ref Expression
hsmexlem.f 𝐹 = OrdIso( E , 𝐵)
hsmexlem.g 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
Assertion
Ref Expression
hsmexlem3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐷 × 𝐶)))
Distinct variable groups:   𝐴,𝑎   𝐶,𝑎
Allowed substitution hints:   𝐵(𝑎)   𝐷(𝑎)   𝐹(𝑎)   𝐺(𝑎)

Proof of Theorem hsmexlem3
StepHypRef Expression
1 wdomref 8518 . . . . 5 (𝐶 ∈ On → 𝐶* 𝐶)
2 xpwdomg 8531 . . . . 5 ((𝐴* 𝐷𝐶* 𝐶) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
31, 2sylan2 490 . . . 4 ((𝐴* 𝐷𝐶 ∈ On) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
4 wdompwdom 8524 . . . 4 ((𝐴 × 𝐶) ≼* (𝐷 × 𝐶) → 𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶))
5 harword 8511 . . . 4 (𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
63, 4, 53syl 18 . . 3 ((𝐴* 𝐷𝐶 ∈ On) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
76adantr 480 . 2 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
8 relwdom 8512 . . . . . 6 Rel ≼*
98brrelexi 5192 . . . . 5 (𝐴* 𝐷𝐴 ∈ V)
109adantr 480 . . . 4 ((𝐴* 𝐷𝐶 ∈ On) → 𝐴 ∈ V)
1110adantr 480 . . 3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐴 ∈ V)
12 simplr 807 . . 3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐶 ∈ On)
13 simpr 476 . . 3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶))
14 hsmexlem.f . . . 4 𝐹 = OrdIso( E , 𝐵)
15 hsmexlem.g . . . 4 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
1614, 15hsmexlem2 9287 . . 3 ((𝐴 ∈ V ∧ 𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
1711, 12, 13, 16syl3anc 1366 . 2 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
187, 17sseldd 3637 1 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐷 × 𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ⊆ wss 3607  𝒫 cpw 4191  ∪ ciun 4552   class class class wbr 4685   E cep 5057   × cxp 5141  dom cdm 5143  Oncon0 5761  ‘cfv 5926   ≼ cdom 7995  OrdIsocoi 8455  harchar 8502   ≼* cwdom 8503 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-smo 7488  df-recs 7513  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-oi 8456  df-har 8504  df-wdom 8505 This theorem is referenced by:  hsmexlem4  9289  hsmexlem5  9290
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