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Theorem hsmexlem2 9200
Description: Lemma for hsmex 9205. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9348 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 4145 . . . . . 6 (𝐵 ∈ 𝒫 On → 𝐵 ⊆ On)
21adantr 481 . . . . 5 ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝐵 ⊆ On)
32ralimi 2947 . . . 4 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → ∀𝑎𝐴 𝐵 ⊆ On)
4 iunss 4532 . . . 4 ( 𝑎𝐴 𝐵 ⊆ On ↔ ∀𝑎𝐴 𝐵 ⊆ On)
53, 4sylibr 224 . . 3 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝑎𝐴 𝐵 ⊆ On)
653ad2ant3 1082 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵 ⊆ On)
7 xpexg 6920 . . . 4 ((𝐴𝑉𝐶 ∈ On) → (𝐴 × 𝐶) ∈ V)
873adant3 1079 . . 3 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐴 × 𝐶) ∈ V)
9 nfv 1840 . . . . . . . . 9 𝑎 𝐶 ∈ On
10 nfra1 2936 . . . . . . . . 9 𝑎𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)
119, 10nfan 1825 . . . . . . . 8 𝑎(𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶))
12 rsp 2924 . . . . . . . . 9 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑎𝐴 → (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)))
13 onelss 5730 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (dom 𝐹𝐶 → dom 𝐹𝐶))
1413imp 445 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ dom 𝐹𝐶) → dom 𝐹𝐶)
1514adantrl 751 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐹𝐶)
16153adant3 1079 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → dom 𝐹𝐶)
17 hsmexlem.f . . . . . . . . . . . . . . . . . . 19 𝐹 = OrdIso( E , 𝐵)
1817oismo 8396 . . . . . . . . . . . . . . . . . 18 (𝐵 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
191, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ 𝒫 On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2019ad2antrl 763 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2120simprd 479 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → ran 𝐹 = 𝐵)
2217oif 8386 . . . . . . . . . . . . . . 15 𝐹:dom 𝐹𝐵
2321, 22jctil 559 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
24 dffo2 6081 . . . . . . . . . . . . . 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 6335 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
2726simprbi 480 . . . . . . . . . . . . 13 (𝐹:dom 𝐹onto𝐵 → ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
28 rsp 2924 . . . . . . . . . . . . 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 1258 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
31 ssrexv 3651 . . . . . . . . . . 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 1261 . . . . . . . . 9 (𝐶 ∈ On → ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3412, 33sylan9r 689 . . . . . . . 8 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑎𝐴 → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3511, 34reximdai 3007 . . . . . . 7 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
36353adant1 1077 . . . . . 6 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
37 nfv 1840 . . . . . . 7 𝑑𝑒𝐶 𝑏 = (𝐹𝑒)
38 nfcv 2761 . . . . . . . 8 𝑎𝐶
39 nfcv 2761 . . . . . . . . . . 11 𝑎 E
40 nfcsb1v 3534 . . . . . . . . . . 11 𝑎𝑑 / 𝑎𝐵
4139, 40nfoi 8370 . . . . . . . . . 10 𝑎OrdIso( E , 𝑑 / 𝑎𝐵)
42 nfcv 2761 . . . . . . . . . 10 𝑎𝑒
4341, 42nffv 6160 . . . . . . . . 9 𝑎(OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4443nfeq2 2776 . . . . . . . 8 𝑎 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4538, 44nfrex 3002 . . . . . . 7 𝑎𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
46 csbeq1a 3527 . . . . . . . . . . . 12 (𝑎 = 𝑑𝐵 = 𝑑 / 𝑎𝐵)
47 oieq2 8369 . . . . . . . . . . . 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 2667 . . . . . . . . . 10 (𝑎 = 𝑑𝐹 = OrdIso( E , 𝑑 / 𝑎𝐵))
5049fveq1d 6155 . . . . . . . . 9 (𝑎 = 𝑑 → (𝐹𝑒) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5150eqeq2d 2631 . . . . . . . 8 (𝑎 = 𝑑 → (𝑏 = (𝐹𝑒) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5251rexbidv 3046 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5337, 45, 52cbvrex 3159 . . . . . 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 4495 . . . . 5 (𝑏 𝑎𝐴 𝐵 ↔ ∃𝑎𝐴 𝑏𝐵)
56 vex 3192 . . . . . . . . . . 11 𝑑 ∈ V
57 vex 3192 . . . . . . . . . . 11 𝑒 ∈ V
5856, 57op1std 7130 . . . . . . . . . 10 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) = 𝑑)
5958csbeq1d 3525 . . . . . . . . 9 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) / 𝑎𝐵 = 𝑑 / 𝑎𝐵)
60 oieq2 8369 . . . . . . . . 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 7131 . . . . . . . 8 (𝑐 = ⟨𝑑, 𝑒⟩ → (2nd𝑐) = 𝑒)
6361, 62fveq12d 6159 . . . . . . 7 (𝑐 = ⟨𝑑, 𝑒⟩ → (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6463eqeq2d 2631 . . . . . 6 (𝑐 = ⟨𝑑, 𝑒⟩ → (𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
6564rexxp 5229 . . . . 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 445 . . 3 (((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) ∧ 𝑏 𝑎𝐴 𝐵) → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)))
688, 67wdomd 8437 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵* (𝐴 × 𝐶))
69 hsmexlem.g . . 3 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
7069hsmexlem1 9199 . 2 (( 𝑎𝐴 𝐵 ⊆ On ∧ 𝑎𝐴 𝐵* (𝐴 × 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
716, 68, 70syl2anc 692 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3189  csb 3518  wss 3559  𝒫 cpw 4135  cop 4159   ciun 4490   class class class wbr 4618   E cep 4988   × cxp 5077  dom cdm 5079  ran crn 5080  Oncon0 5687  wf 5848  ontowfo 5850  cfv 5852  1st c1st 7118  2nd c2nd 7119  Smo wsmo 7394  OrdIsocoi 8365  harchar 8412  * cwdom 8413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-se 5039  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-isom 5861  df-riota 6571  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-smo 7395  df-recs 7420  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-oi 8366  df-har 8414  df-wdom 8415
This theorem is referenced by:  hsmexlem3  9201
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