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Theorem hsmexlem3 10410
Description: Lemma for hsmex 10414. 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 9554 . . . . 5 (𝐶 ∈ On → 𝐶* 𝐶)
2 xpwdomg 9567 . . . . 5 ((𝐴* 𝐷𝐶* 𝐶) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
31, 2sylan2 594 . . . 4 ((𝐴* 𝐷𝐶 ∈ On) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
4 wdompwdom 9560 . . . 4 ((𝐴 × 𝐶) ≼* (𝐷 × 𝐶) → 𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶))
5 harword 9545 . . . 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 482 . 2 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (har‘𝒫 (𝐴 × 𝐶)) ⊆ (har‘𝒫 (𝐷 × 𝐶)))
8 relwdom 9548 . . . . . 6 Rel ≼*
98brrelex1i 5727 . . . . 5 (𝐴* 𝐷𝐴 ∈ V)
109adantr 482 . . . 4 ((𝐴* 𝐷𝐶 ∈ On) → 𝐴 ∈ V)
1110adantr 482 . . 3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐴 ∈ V)
12 simplr 768 . . 3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐶 ∈ On)
13 simpr 486 . . 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 10409 . . 3 ((𝐴 ∈ V ∧ 𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
1711, 12, 13, 16syl3anc 1372 . 2 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
187, 17sseldd 3981 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 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  wss 3946  𝒫 cpw 4598   ciun 4993   class class class wbr 5144   E cep 5575   × cxp 5670  dom cdm 5672  Oncon0 6356  cfv 6535  cdom 8925  OrdIsocoi 9491  harchar 9538  * cwdom 9546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6292  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-isom 6544  df-riota 7352  df-ov 7399  df-1st 7962  df-2nd 7963  df-frecs 8253  df-wrecs 8284  df-smo 8333  df-recs 8358  df-en 8928  df-dom 8929  df-sdom 8930  df-oi 9492  df-har 9539  df-wdom 9547
This theorem is referenced by:  hsmexlem4  10411  hsmexlem5  10412
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