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Theorem hsmexlem3 10465
Description: Lemma for hsmex 10469. 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 9609 . . . . 5 (𝐶 ∈ On → 𝐶* 𝐶)
2 xpwdomg 9622 . . . . 5 ((𝐴* 𝐷𝐶* 𝐶) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
31, 2sylan2 593 . . . 4 ((𝐴* 𝐷𝐶 ∈ On) → (𝐴 × 𝐶) ≼* (𝐷 × 𝐶))
4 wdompwdom 9615 . . . 4 ((𝐴 × 𝐶) ≼* (𝐷 × 𝐶) → 𝒫 (𝐴 × 𝐶) ≼ 𝒫 (𝐷 × 𝐶))
5 harword 9600 . . . 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 9603 . . . . . 6 Rel ≼*
98brrelex1i 5744 . . . . 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 769 . . 3 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐶 ∈ On)
13 simpr 484 . . 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 10464 . . 3 ((𝐴 ∈ V ∧ 𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
1711, 12, 13, 16syl3anc 1370 . 2 (((𝐴* 𝐷𝐶 ∈ On) ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
187, 17sseldd 3995 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 395   = wceq 1536  wcel 2105  wral 3058  Vcvv 3477  wss 3962  𝒫 cpw 4604   ciun 4995   class class class wbr 5147   E cep 5587   × cxp 5686  dom cdm 5688  Oncon0 6385  cfv 6562  cdom 8981  OrdIsocoi 9546  harchar 9593  * cwdom 9601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-smo 8384  df-recs 8409  df-en 8984  df-dom 8985  df-sdom 8986  df-oi 9547  df-har 9594  df-wdom 9602
This theorem is referenced by:  hsmexlem4  10466  hsmexlem5  10467
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