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Theorem hsmexlem3 9452
 Description: Lemma for hsmex 9456. 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 8633 . . . . 5 (𝐶 ∈ On → 𝐶* 𝐶)
2 xpwdomg 8646 . . . . 5 ((𝐴* 𝐷𝐶* 𝐶) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
31, 2sylan2 580 . . . 4 ((𝐴* 𝐷𝐶 ∈ On) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
4 wdompwdom 8639 . . . 4 ((𝐴 × 𝐶) ≼* (𝐷 × 𝐶) → 𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶))
5 harword 8626 . . . 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 466 . 2 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
8 relwdom 8627 . . . . . 6 Rel ≼*
98brrelexi 5298 . . . . 5 (𝐴* 𝐷𝐴 ∈ V)
109adantr 466 . . . 4 ((𝐴* 𝐷𝐶 ∈ On) → 𝐴 ∈ V)
1110adantr 466 . . 3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐴 ∈ V)
12 simplr 752 . . 3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐶 ∈ On)
13 simpr 471 . . 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 9451 . . 3 ((𝐴 ∈ V ∧ 𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
1711, 12, 13, 16syl3anc 1476 . 2 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
187, 17sseldd 3753 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 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ⊆ wss 3723  𝒫 cpw 4297  ∪ ciun 4654   class class class wbr 4786   E cep 5161   × cxp 5247  dom cdm 5249  Oncon0 5866  ‘cfv 6031   ≼ cdom 8107  OrdIsocoi 8570  harchar 8617   ≼* cwdom 8618 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-smo 7596  df-recs 7621  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-oi 8571  df-har 8619  df-wdom 8620 This theorem is referenced by:  hsmexlem4  9453  hsmexlem5  9454
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