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Theorem hsmexlem2 9502
Description: Lemma for hsmex 9507. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 9650 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 4325 . . . . . 6 (𝐵 ∈ 𝒫 On → 𝐵 ⊆ On)
21adantr 472 . . . . 5 ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝐵 ⊆ On)
32ralimi 3099 . . . 4 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → ∀𝑎𝐴 𝐵 ⊆ On)
4 iunss 4717 . . . 4 ( 𝑎𝐴 𝐵 ⊆ On ↔ ∀𝑎𝐴 𝐵 ⊆ On)
53, 4sylibr 225 . . 3 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → 𝑎𝐴 𝐵 ⊆ On)
653ad2ant3 1165 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵 ⊆ On)
7 xpexg 7158 . . . 4 ((𝐴𝑉𝐶 ∈ On) → (𝐴 × 𝐶) ∈ V)
873adant3 1162 . . 3 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐴 × 𝐶) ∈ V)
9 nfv 2009 . . . . . . . . 9 𝑎 𝐶 ∈ On
10 nfra1 3088 . . . . . . . . 9 𝑎𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)
119, 10nfan 1998 . . . . . . . 8 𝑎(𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶))
12 rsp 3076 . . . . . . . . 9 (∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑎𝐴 → (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)))
13 onelss 5950 . . . . . . . . . . . . . 14 (𝐶 ∈ On → (dom 𝐹𝐶 → dom 𝐹𝐶))
1413imp 395 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ dom 𝐹𝐶) → dom 𝐹𝐶)
1514adantrl 707 . . . . . . . . . . . 12 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → dom 𝐹𝐶)
16153adant3 1162 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → dom 𝐹𝐶)
17 hsmexlem.f . . . . . . . . . . . . . . . . . . 19 𝐹 = OrdIso( E , 𝐵)
1817oismo 8652 . . . . . . . . . . . . . . . . . 18 (𝐵 ⊆ On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
191, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝐵 ∈ 𝒫 On → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2019ad2antrl 719 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (Smo 𝐹 ∧ ran 𝐹 = 𝐵))
2120simprd 489 . . . . . . . . . . . . . . 15 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → ran 𝐹 = 𝐵)
2217oif 8642 . . . . . . . . . . . . . . 15 𝐹:dom 𝐹𝐵
2321, 22jctil 515 . . . . . . . . . . . . . 14 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
24 dffo2 6302 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ran 𝐹 = 𝐵))
2523, 24sylibr 225 . . . . . . . . . . . . 13 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝐹:dom 𝐹onto𝐵)
26 dffo3 6564 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹onto𝐵 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒)))
2726simprbi 490 . . . . . . . . . . . . 13 (𝐹:dom 𝐹onto𝐵 → ∀𝑏𝐵𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
28 rsp 3076 . . . . . . . . . . . . 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 1145 . . . . . . . . . . 11 ((𝐶 ∈ On ∧ (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) ∧ 𝑏𝐵) → ∃𝑒 ∈ dom 𝐹 𝑏 = (𝐹𝑒))
31 ssrexv 3827 . . . . . . . . . . 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 1148 . . . . . . . . 9 (𝐶 ∈ On → ((𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶) → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3412, 33sylan9r 504 . . . . . . . 8 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑎𝐴 → (𝑏𝐵 → ∃𝑒𝐶 𝑏 = (𝐹𝑒))))
3511, 34reximdai 3158 . . . . . . 7 ((𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
36353adant1 1160 . . . . . 6 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒)))
37 nfv 2009 . . . . . . 7 𝑑𝑒𝐶 𝑏 = (𝐹𝑒)
38 nfcv 2907 . . . . . . . 8 𝑎𝐶
39 nfcv 2907 . . . . . . . . . . 11 𝑎 E
40 nfcsb1v 3707 . . . . . . . . . . 11 𝑎𝑑 / 𝑎𝐵
4139, 40nfoi 8626 . . . . . . . . . 10 𝑎OrdIso( E , 𝑑 / 𝑎𝐵)
42 nfcv 2907 . . . . . . . . . 10 𝑎𝑒
4341, 42nffv 6385 . . . . . . . . 9 𝑎(OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4443nfeq2 2923 . . . . . . . 8 𝑎 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
4538, 44nfrex 3153 . . . . . . 7 𝑎𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)
46 csbeq1a 3700 . . . . . . . . . . . 12 (𝑎 = 𝑑𝐵 = 𝑑 / 𝑎𝐵)
47 oieq2 8625 . . . . . . . . . . . 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 2811 . . . . . . . . . 10 (𝑎 = 𝑑𝐹 = OrdIso( E , 𝑑 / 𝑎𝐵))
5049fveq1d 6377 . . . . . . . . 9 (𝑎 = 𝑑 → (𝐹𝑒) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5150eqeq2d 2775 . . . . . . . 8 (𝑎 = 𝑑 → (𝑏 = (𝐹𝑒) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5251rexbidv 3199 . . . . . . 7 (𝑎 = 𝑑 → (∃𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
5337, 45, 52cbvrex 3316 . . . . . 6 (∃𝑎𝐴𝑒𝐶 𝑏 = (𝐹𝑒) ↔ ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
5436, 53syl6ib 242 . . . . 5 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (∃𝑎𝐴 𝑏𝐵 → ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
55 eliun 4680 . . . . 5 (𝑏 𝑎𝐴 𝐵 ↔ ∃𝑎𝐴 𝑏𝐵)
56 vex 3353 . . . . . . . . . . 11 𝑑 ∈ V
57 vex 3353 . . . . . . . . . . 11 𝑒 ∈ V
5856, 57op1std 7376 . . . . . . . . . 10 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) = 𝑑)
5958csbeq1d 3698 . . . . . . . . 9 (𝑐 = ⟨𝑑, 𝑒⟩ → (1st𝑐) / 𝑎𝐵 = 𝑑 / 𝑎𝐵)
60 oieq2 8625 . . . . . . . . 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 7377 . . . . . . . 8 (𝑐 = ⟨𝑑, 𝑒⟩ → (2nd𝑐) = 𝑒)
6361, 62fveq12d 6382 . . . . . . 7 (𝑐 = ⟨𝑑, 𝑒⟩ → (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6463eqeq2d 2775 . . . . . 6 (𝑐 = ⟨𝑑, 𝑒⟩ → (𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒)))
6564rexxp 5433 . . . . 5 (∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)) ↔ ∃𝑑𝐴𝑒𝐶 𝑏 = (OrdIso( E , 𝑑 / 𝑎𝐵)‘𝑒))
6654, 55, 653imtr4g 287 . . . 4 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → (𝑏 𝑎𝐴 𝐵 → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐))))
6766imp 395 . . 3 (((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) ∧ 𝑏 𝑎𝐴 𝐵) → ∃𝑐 ∈ (𝐴 × 𝐶)𝑏 = (OrdIso( E , (1st𝑐) / 𝑎𝐵)‘(2nd𝑐)))
688, 67wdomd 8693 . 2 ((𝐴𝑉𝐶 ∈ On ∧ ∀𝑎𝐴 (𝐵 ∈ 𝒫 On ∧ dom 𝐹𝐶)) → 𝑎𝐴 𝐵* (𝐴 × 𝐶))
69 hsmexlem.g . . 3 𝐺 = OrdIso( E , 𝑎𝐴 𝐵)
7069hsmexlem1 9501 . 2 (( 𝑎𝐴 𝐵 ⊆ On ∧ 𝑎𝐴 𝐵* (𝐴 × 𝐶)) → dom 𝐺 ∈ (har‘𝒫 (𝐴 × 𝐶)))
716, 68, 70syl2anc 579 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 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  csb 3691  wss 3732  𝒫 cpw 4315  cop 4340   ciun 4676   class class class wbr 4809   E cep 5189   × cxp 5275  dom cdm 5277  ran crn 5278  Oncon0 5908  wf 6064  ontowfo 6066  cfv 6068  1st c1st 7364  2nd c2nd 7365  Smo wsmo 7646  OrdIsocoi 8621  harchar 8668  * cwdom 8669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-smo 7647  df-recs 7672  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-oi 8622  df-har 8670  df-wdom 8671
This theorem is referenced by:  hsmexlem3  9503
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