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Theorem hsmexlem2 9842
Description: Lemma for hsmex 9847. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9990 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 4509 . . . . . 6 (𝐵 ∈ 𝒫 On → 𝐵 ⊆ On)
21adantr 484 . . . . 5 ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝐵 ⊆ On)
32ralimi 3131 . . . 4 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → ∀𝑎𝐴 𝐵 ⊆ On)
4 iunss 4935 . . . 4 ( 𝑎𝐴 𝐵 ⊆ On ↔ ∀𝑎𝐴 𝐵 ⊆ On)
53, 4sylibr 237 . . 3 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝑎𝐴 𝐵 ⊆ On)
653ad2ant3 1132 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵 ⊆ On)
7 xpexg 7457 . . . 4 ((𝐴𝑉𝐶 ∈ On) → (𝐴 × 𝐶) ∈ V)
873adant3 1129 . . 3 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐴 × 𝐶) ∈ V)
9 nfv 1915 . . . . . . . . 9 𝑎 𝐶 ∈ On
10 nfra1 3186 . . . . . . . . 9 𝑎𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)
119, 10nfan 1900 . . . . . . . 8 𝑎(𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶))
12 rsp 3173 . . . . . . . . 9 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑎𝐴 → (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)))
13 onelss 6205 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (dom 𝐹𝐶 → dom 𝐹𝐶))
1413imp 410 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ dom 𝐹𝐶) → dom 𝐹𝐶)
1514adantrl 715 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐹𝐶)
16153adant3 1129 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → dom 𝐹𝐶)
17 hsmexlem.f . . . . . . . . . . . . . . . . . . 19 𝐹 = OrdIso( E , 𝐵)
1817oismo 8992 . . . . . . . . . . . . . . . . . 18 (𝐵 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
191, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ 𝒫 On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2019ad2antrl 727 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2120simprd 499 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → ran 𝐹 = 𝐵)
2217oif 8982 . . . . . . . . . . . . . . 15 𝐹:dom 𝐹𝐵
2321, 22jctil 523 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
24 dffo2 6573 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
2523, 24sylibr 237 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐹:dom 𝐹onto𝐵)
26 dffo3 6849 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
2726simprbi 500 . . . . . . . . . . . . 13 (𝐹:dom 𝐹onto𝐵 → ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
28 rsp 3173 . . . . . . . . . . . . 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 1114 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
31 ssrexv 3985 . . . . . . . . . . 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 1116 . . . . . . . . 9 (𝐶 ∈ On → ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3412, 33sylan9r 512 . . . . . . . 8 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑎𝐴 → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3511, 34reximdai 3273 . . . . . . 7 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
36353adant1 1127 . . . . . 6 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
37 nfv 1915 . . . . . . 7 𝑑𝑒𝐶 𝑏 = (𝐹𝑒)
38 nfcv 2958 . . . . . . . 8 𝑎𝐶
39 nfcv 2958 . . . . . . . . . . 11 𝑎 E
40 nfcsb1v 3855 . . . . . . . . . . 11 𝑎𝑑 / 𝑎𝐵
4139, 40nfoi 8966 . . . . . . . . . 10 𝑎OrdIso( E , 𝑑 / 𝑎𝐵)
42 nfcv 2958 . . . . . . . . . 10 𝑎𝑒
4341, 42nffv 6659 . . . . . . . . 9 𝑎(OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4443nfeq2 2975 . . . . . . . 8 𝑎 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4538, 44nfrex 3271 . . . . . . 7 𝑎𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
46 csbeq1a 3845 . . . . . . . . . . . 12 (𝑎 = 𝑑𝐵 = 𝑑 / 𝑎𝐵)
47 oieq2 8965 . . . . . . . . . . . 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 2848 . . . . . . . . . 10 (𝑎 = 𝑑𝐹 = OrdIso( E , 𝑑 / 𝑎𝐵))
5049fveq1d 6651 . . . . . . . . 9 (𝑎 = 𝑑 → (𝐹𝑒) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5150eqeq2d 2812 . . . . . . . 8 (𝑎 = 𝑑 → (𝑏 = (𝐹𝑒) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5251rexbidv 3259 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5337, 45, 52cbvrexw 3391 . . . . . 6 (∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5436, 53syl6ib 254 . . . . 5 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
55 eliun 4888 . . . . 5 (𝑏 𝑎𝐴 𝐵 ↔ ∃𝑎𝐴 𝑏𝐵)
56 vex 3447 . . . . . . . . . . 11 𝑑 ∈ V
57 vex 3447 . . . . . . . . . . 11 𝑒 ∈ V
5856, 57op1std 7685 . . . . . . . . . 10 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) = 𝑑)
5958csbeq1d 3835 . . . . . . . . 9 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) / 𝑎𝐵 = 𝑑 / 𝑎𝐵)
60 oieq2 8965 . . . . . . . . 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 7686 . . . . . . . 8 (𝑐 = ⟨𝑑, 𝑒⟩ → (2nd𝑐) = 𝑒)
6361, 62fveq12d 6656 . . . . . . 7 (𝑐 = ⟨𝑑, 𝑒⟩ → (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6463eqeq2d 2812 . . . . . 6 (𝑐 = ⟨𝑑, 𝑒⟩ → (𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
6564rexxp 5681 . . . . 5 (∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6654, 55, 653imtr4g 299 . . . 4 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑏 𝑎𝐴 𝐵 → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐))))
6766imp 410 . . 3 (((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) ∧ 𝑏 𝑎𝐴 𝐵) → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)))
688, 67wdomd 9033 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵* (𝐴 × 𝐶))
69 hsmexlem.g . . 3 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
7069hsmexlem1 9841 . 2 (( 𝑎𝐴 𝐵 ⊆ On ∧ 𝑎𝐴 𝐵* (𝐴 × 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
716, 68, 70syl2anc 587 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 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  wrex 3110  Vcvv 3444  csb 3831  wss 3884  𝒫 cpw 4500  cop 4534   ciun 4884   class class class wbr 5033   E cep 5432   × cxp 5521  dom cdm 5523  ran crn 5524  Oncon0 6163  wf 6324  ontowfo 6326  cfv 6328  1st c1st 7673  2nd c2nd 7674  Smo wsmo 7969  OrdIsocoi 8961  harchar 9008  * cwdom 9016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-smo 7970  df-recs 7995  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-oi 8962  df-har 9009  df-wdom 9017
This theorem is referenced by:  hsmexlem3  9843
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