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Theorem hsmexlem2 9287
Description: Lemma for hsmex 9292. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9435 but use of order types allows to canonically choose the sub-bijections, removing the choice requirement. (Contributed by Stefan O'Rear, 14-Feb-2015.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by AV, 18-Sep-2021.)
Hypotheses
Ref Expression
hsmexlem.f 𝐹 = OrdIso( E , 𝐵)
hsmexlem.g 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
Assertion
Ref Expression
hsmexlem2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
Distinct variable groups:   𝐴,𝑎   𝐶,𝑎
Allowed substitution hints:   𝐵(𝑎)   𝐹(𝑎)   𝐺(𝑎)   𝑉(𝑎)

Proof of Theorem hsmexlem2
Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 4201 . . . . . 6 (𝐵 ∈ 𝒫 On → 𝐵 ⊆ On)
21adantr 480 . . . . 5 ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝐵 ⊆ On)
32ralimi 2981 . . . 4 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → ∀𝑎𝐴 𝐵 ⊆ On)
4 iunss 4593 . . . 4 ( 𝑎𝐴 𝐵 ⊆ On ↔ ∀𝑎𝐴 𝐵 ⊆ On)
53, 4sylibr 224 . . 3 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝑎𝐴 𝐵 ⊆ On)
653ad2ant3 1104 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵 ⊆ On)
7 xpexg 7002 . . . 4 ((𝐴𝑉𝐶 ∈ On) → (𝐴 × 𝐶) ∈ V)
873adant3 1101 . . 3 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐴 × 𝐶) ∈ V)
9 nfv 1883 . . . . . . . . 9 𝑎 𝐶 ∈ On
10 nfra1 2970 . . . . . . . . 9 𝑎𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)
119, 10nfan 1868 . . . . . . . 8 𝑎(𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶))
12 rsp 2958 . . . . . . . . 9 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑎𝐴 → (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)))
13 onelss 5804 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (dom 𝐹𝐶 → dom 𝐹𝐶))
1413imp 444 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ dom 𝐹𝐶) → dom 𝐹𝐶)
1514adantrl 752 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐹𝐶)
16153adant3 1101 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → dom 𝐹𝐶)
17 hsmexlem.f . . . . . . . . . . . . . . . . . . 19 𝐹 = OrdIso( E , 𝐵)
1817oismo 8486 . . . . . . . . . . . . . . . . . 18 (𝐵 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
191, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ 𝒫 On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2019ad2antrl 764 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2120simprd 478 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → ran 𝐹 = 𝐵)
2217oif 8476 . . . . . . . . . . . . . . 15 𝐹:dom 𝐹𝐵
2321, 22jctil 559 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
24 dffo2 6157 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
2523, 24sylibr 224 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐹:dom 𝐹onto𝐵)
26 dffo3 6414 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
2726simprbi 479 . . . . . . . . . . . . 13 (𝐹:dom 𝐹onto𝐵 → ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
28 rsp 2958 . . . . . . . . . . . . 13 (∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒) → (𝑏𝐵 → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
2925, 27, 283syl 18 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑏𝐵 → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
30293impia 1280 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
31 ssrexv 3700 . . . . . . . . . . 11 (dom 𝐹𝐶 → (∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒) → ∃𝑒𝐶 𝑏 = (𝐹𝑒)))
3216, 30, 31sylc 65 . . . . . . . . . 10 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → ∃𝑒𝐶 𝑏 = (𝐹𝑒))
33323exp 1283 . . . . . . . . 9 (𝐶 ∈ On → ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3412, 33sylan9r 691 . . . . . . . 8 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑎𝐴 → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3511, 34reximdai 3041 . . . . . . 7 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
36353adant1 1099 . . . . . 6 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
37 nfv 1883 . . . . . . 7 𝑑𝑒𝐶 𝑏 = (𝐹𝑒)
38 nfcv 2793 . . . . . . . 8 𝑎𝐶
39 nfcv 2793 . . . . . . . . . . 11 𝑎 E
40 nfcsb1v 3582 . . . . . . . . . . 11 𝑎𝑑 / 𝑎𝐵
4139, 40nfoi 8460 . . . . . . . . . 10 𝑎OrdIso( E , 𝑑 / 𝑎𝐵)
42 nfcv 2793 . . . . . . . . . 10 𝑎𝑒
4341, 42nffv 6236 . . . . . . . . 9 𝑎(OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4443nfeq2 2809 . . . . . . . 8 𝑎 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4538, 44nfrex 3036 . . . . . . 7 𝑎𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
46 csbeq1a 3575 . . . . . . . . . . . 12 (𝑎 = 𝑑𝐵 = 𝑑 / 𝑎𝐵)
47 oieq2 8459 . . . . . . . . . . . 12 (𝐵 = 𝑑 / 𝑎𝐵 → OrdIso( E , 𝐵) = OrdIso( E , 𝑑 / 𝑎𝐵))
4846, 47syl 17 . . . . . . . . . . 11 (𝑎 = 𝑑 → OrdIso( E , 𝐵) = OrdIso( E , 𝑑 / 𝑎𝐵))
4917, 48syl5eq 2697 . . . . . . . . . 10 (𝑎 = 𝑑𝐹 = OrdIso( E , 𝑑 / 𝑎𝐵))
5049fveq1d 6231 . . . . . . . . 9 (𝑎 = 𝑑 → (𝐹𝑒) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5150eqeq2d 2661 . . . . . . . 8 (𝑎 = 𝑑 → (𝑏 = (𝐹𝑒) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5251rexbidv 3081 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5337, 45, 52cbvrex 3198 . . . . . 6 (∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5436, 53syl6ib 241 . . . . 5 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
55 eliun 4556 . . . . 5 (𝑏 𝑎𝐴 𝐵 ↔ ∃𝑎𝐴 𝑏𝐵)
56 vex 3234 . . . . . . . . . . 11 𝑑 ∈ V
57 vex 3234 . . . . . . . . . . 11 𝑒 ∈ V
5856, 57op1std 7220 . . . . . . . . . 10 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) = 𝑑)
5958csbeq1d 3573 . . . . . . . . 9 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) / 𝑎𝐵 = 𝑑 / 𝑎𝐵)
60 oieq2 8459 . . . . . . . . 9 ((1st𝑐) / 𝑎𝐵 = 𝑑 / 𝑎𝐵 → OrdIso( E , (1st𝑐) / 𝑎𝐵) = OrdIso( E , 𝑑 / 𝑎𝐵))
6159, 60syl 17 . . . . . . . 8 (𝑐 = ⟨𝑑, 𝑒⟩ → OrdIso( E , (1st𝑐) / 𝑎𝐵) = OrdIso( E , 𝑑 / 𝑎𝐵))
6256, 57op2ndd 7221 . . . . . . . 8 (𝑐 = ⟨𝑑, 𝑒⟩ → (2nd𝑐) = 𝑒)
6361, 62fveq12d 6235 . . . . . . 7 (𝑐 = ⟨𝑑, 𝑒⟩ → (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6463eqeq2d 2661 . . . . . 6 (𝑐 = ⟨𝑑, 𝑒⟩ → (𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
6564rexxp 5297 . . . . 5 (∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6654, 55, 653imtr4g 285 . . . 4 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑏 𝑎𝐴 𝐵 → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐))))
6766imp 444 . . 3 (((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) ∧ 𝑏 𝑎𝐴 𝐵) → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)))
688, 67wdomd 8527 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵* (𝐴 × 𝐶))
69 hsmexlem.g . . 3 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
7069hsmexlem1 9286 . 2 (( 𝑎𝐴 𝐵 ⊆ On ∧ 𝑎𝐴 𝐵* (𝐴 × 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
716, 68, 70syl2anc 694 1 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  Vcvv 3231  csb 3566  wss 3607  𝒫 cpw 4191  cop 4216   ciun 4552   class class class wbr 4685   E cep 5057   × cxp 5141  dom cdm 5143  ran crn 5144  Oncon0 5761  wf 5922  ontowfo 5924  cfv 5926  1st c1st 7208  2nd c2nd 7209  Smo wsmo 7487  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:  hsmexlem3  9288
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