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Theorem hsmexlem2 10426
Description: Lemma for hsmex 10431. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 10574 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 , 𝐡)
hsmexlem.g 𝐺 = OrdIso( E , βˆͺ π‘Ž ∈ 𝐴 𝐡)
Assertion
Ref Expression
hsmexlem2 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ dom 𝐺 ∈ (harβ€˜π’« (𝐴 Γ— 𝐢)))
Distinct variable groups:   𝐴,π‘Ž   𝐢,π‘Ž
Allowed substitution hints:   𝐡(π‘Ž)   𝐹(π‘Ž)   𝐺(π‘Ž)   𝑉(π‘Ž)
Proof of Theorem hsmexlem2
Dummy variables 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elpwi 4609 . . . . . 6 (𝐡 ∈ 𝒫 On β†’ 𝐡 βŠ† On)
21adantr 480 . . . . 5 ((𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) β†’ 𝐡 βŠ† On)
32ralimi 3082 . . . 4 (βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) β†’ βˆ€π‘Ž ∈ 𝐴 𝐡 βŠ† On)
4 iunss 5048 . . . 4 (βˆͺ π‘Ž ∈ 𝐴 𝐡 βŠ† On ↔ βˆ€π‘Ž ∈ 𝐴 𝐡 βŠ† On)
53, 4sylibr 233 . . 3 (βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) β†’ βˆͺ π‘Ž ∈ 𝐴 𝐡 βŠ† On)
653ad2ant3 1134 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ βˆͺ π‘Ž ∈ 𝐴 𝐡 βŠ† On)
7 xpexg 7741 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On) β†’ (𝐴 Γ— 𝐢) ∈ V)
873adant3 1131 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (𝐴 Γ— 𝐢) ∈ V)
9 nfv 1916 . . . . . . . . 9 β„²π‘Ž 𝐢 ∈ On
10 nfra1 3280 . . . . . . . . 9 β„²π‘Žβˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)
119, 10nfan 1901 . . . . . . . 8 β„²π‘Ž(𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢))
12 rsp 3243 . . . . . . . . 9 (βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) β†’ (π‘Ž ∈ 𝐴 β†’ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)))
13 onelss 6406 . . . . . . . . . . . . . 14 (𝐢 ∈ On β†’ (dom 𝐹 ∈ 𝐢 β†’ dom 𝐹 βŠ† 𝐢))
1413imp 406 . . . . . . . . . . . . 13 ((𝐢 ∈ On ∧ dom 𝐹 ∈ 𝐢) β†’ dom 𝐹 βŠ† 𝐢)
1514adantrl 713 . . . . . . . . . . . 12 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ dom 𝐹 βŠ† 𝐢)
16153adant3 1131 . . . . . . . . . . 11 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) ∧ 𝑏 ∈ 𝐡) β†’ dom 𝐹 βŠ† 𝐢)
17 hsmexlem.f . . . . . . . . . . . . . . . . . . 19 𝐹 = OrdIso( E , 𝐡)
1817oismo 9539 . . . . . . . . . . . . . . . . . 18 (𝐡 βŠ† On β†’ (Smo 𝐹 ∧ ran 𝐹 = 𝐡))
191, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝐡 ∈ 𝒫 On β†’ (Smo 𝐹 ∧ ran 𝐹 = 𝐡))
2019ad2antrl 725 . . . . . . . . . . . . . . . 16 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (Smo 𝐹 ∧ ran 𝐹 = 𝐡))
2120simprd 495 . . . . . . . . . . . . . . 15 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ ran 𝐹 = 𝐡)
2217oif 9529 . . . . . . . . . . . . . . 15 𝐹:dom 𝐹⟢𝐡
2321, 22jctil 519 . . . . . . . . . . . . . 14 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (𝐹:dom 𝐹⟢𝐡 ∧ ran 𝐹 = 𝐡))
24 dffo2 6809 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹–onto→𝐡 ↔ (𝐹:dom 𝐹⟢𝐡 ∧ ran 𝐹 = 𝐡))
2523, 24sylibr 233 . . . . . . . . . . . . 13 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ 𝐹:dom 𝐹–onto→𝐡)
26 dffo3 7103 . . . . . . . . . . . . . 14 (𝐹:dom 𝐹–onto→𝐡 ↔ (𝐹:dom 𝐹⟢𝐡 ∧ βˆ€π‘ ∈ 𝐡 βˆƒπ‘’ ∈ dom 𝐹 𝑏 = (πΉβ€˜π‘’)))
2726simprbi 496 . . . . . . . . . . . . 13 (𝐹:dom 𝐹–onto→𝐡 β†’ βˆ€π‘ ∈ 𝐡 βˆƒπ‘’ ∈ dom 𝐹 𝑏 = (πΉβ€˜π‘’))
28 rsp 3243 . . . . . . . . . . . . 13 (βˆ€π‘ ∈ 𝐡 βˆƒπ‘’ ∈ dom 𝐹 𝑏 = (πΉβ€˜π‘’) β†’ (𝑏 ∈ 𝐡 β†’ βˆƒπ‘’ ∈ dom 𝐹 𝑏 = (πΉβ€˜π‘’)))
2925, 27, 283syl 18 . . . . . . . . . . . 12 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (𝑏 ∈ 𝐡 β†’ βˆƒπ‘’ ∈ dom 𝐹 𝑏 = (πΉβ€˜π‘’)))
30293impia 1116 . . . . . . . . . . 11 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) ∧ 𝑏 ∈ 𝐡) β†’ βˆƒπ‘’ ∈ dom 𝐹 𝑏 = (πΉβ€˜π‘’))
31 ssrexv 4051 . . . . . . . . . . 11 (dom 𝐹 βŠ† 𝐢 β†’ (βˆƒπ‘’ ∈ dom 𝐹 𝑏 = (πΉβ€˜π‘’) β†’ βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’)))
3216, 30, 31sylc 65 . . . . . . . . . 10 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) ∧ 𝑏 ∈ 𝐡) β†’ βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’))
33323exp 1118 . . . . . . . . 9 (𝐢 ∈ On β†’ ((𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) β†’ (𝑏 ∈ 𝐡 β†’ βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’))))
3412, 33sylan9r 508 . . . . . . . 8 ((𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (π‘Ž ∈ 𝐴 β†’ (𝑏 ∈ 𝐡 β†’ βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’))))
3511, 34reximdai 3257 . . . . . . 7 ((𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (βˆƒπ‘Ž ∈ 𝐴 𝑏 ∈ 𝐡 β†’ βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’)))
36353adant1 1129 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (βˆƒπ‘Ž ∈ 𝐴 𝑏 ∈ 𝐡 β†’ βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’)))
37 nfv 1916 . . . . . . 7 β„²π‘‘βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’)
38 nfcv 2902 . . . . . . . 8 β„²π‘ŽπΆ
39 nfcv 2902 . . . . . . . . . . 11 β„²π‘Ž E
40 nfcsb1v 3918 . . . . . . . . . . 11 β„²π‘Žβ¦‹π‘‘ / π‘Žβ¦Œπ΅
4139, 40nfoi 9513 . . . . . . . . . 10 β„²π‘ŽOrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)
42 nfcv 2902 . . . . . . . . . 10 β„²π‘Žπ‘’
4341, 42nffv 6901 . . . . . . . . 9 β„²π‘Ž(OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)
4443nfeq2 2919 . . . . . . . 8 β„²π‘Ž 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)
4538, 44nfrexw 3309 . . . . . . 7 β„²π‘Žβˆƒπ‘’ ∈ 𝐢 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)
46 csbeq1a 3907 . . . . . . . . . . . 12 (π‘Ž = 𝑑 β†’ 𝐡 = ⦋𝑑 / π‘Žβ¦Œπ΅)
47 oieq2 9512 . . . . . . . . . . . 12 (𝐡 = ⦋𝑑 / π‘Žβ¦Œπ΅ β†’ OrdIso( E , 𝐡) = OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅))
4846, 47syl 17 . . . . . . . . . . 11 (π‘Ž = 𝑑 β†’ OrdIso( E , 𝐡) = OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅))
4917, 48eqtrid 2783 . . . . . . . . . 10 (π‘Ž = 𝑑 β†’ 𝐹 = OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅))
5049fveq1d 6893 . . . . . . . . 9 (π‘Ž = 𝑑 β†’ (πΉβ€˜π‘’) = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’))
5150eqeq2d 2742 . . . . . . . 8 (π‘Ž = 𝑑 β†’ (𝑏 = (πΉβ€˜π‘’) ↔ 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)))
5251rexbidv 3177 . . . . . . 7 (π‘Ž = 𝑑 β†’ (βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’) ↔ βˆƒπ‘’ ∈ 𝐢 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)))
5337, 45, 52cbvrexw 3303 . . . . . 6 (βˆƒπ‘Ž ∈ 𝐴 βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’) ↔ βˆƒπ‘‘ ∈ 𝐴 βˆƒπ‘’ ∈ 𝐢 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’))
5436, 53imbitrdi 250 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (βˆƒπ‘Ž ∈ 𝐴 𝑏 ∈ 𝐡 β†’ βˆƒπ‘‘ ∈ 𝐴 βˆƒπ‘’ ∈ 𝐢 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)))
55 eliun 5001 . . . . 5 (𝑏 ∈ βˆͺ π‘Ž ∈ 𝐴 𝐡 ↔ βˆƒπ‘Ž ∈ 𝐴 𝑏 ∈ 𝐡)
56 vex 3477 . . . . . . . . . . 11 𝑑 ∈ V
57 vex 3477 . . . . . . . . . . 11 𝑒 ∈ V
5856, 57op1std 7989 . . . . . . . . . 10 (𝑐 = βŸ¨π‘‘, π‘’βŸ© β†’ (1st β€˜π‘) = 𝑑)
5958csbeq1d 3897 . . . . . . . . 9 (𝑐 = βŸ¨π‘‘, π‘’βŸ© β†’ ⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅ = ⦋𝑑 / π‘Žβ¦Œπ΅)
60 oieq2 9512 . . . . . . . . 9 (⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅ = ⦋𝑑 / π‘Žβ¦Œπ΅ β†’ OrdIso( E , ⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅) = OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅))
6159, 60syl 17 . . . . . . . 8 (𝑐 = βŸ¨π‘‘, π‘’βŸ© β†’ OrdIso( E , ⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅) = OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅))
6256, 57op2ndd 7990 . . . . . . . 8 (𝑐 = βŸ¨π‘‘, π‘’βŸ© β†’ (2nd β€˜π‘) = 𝑒)
6361, 62fveq12d 6898 . . . . . . 7 (𝑐 = βŸ¨π‘‘, π‘’βŸ© β†’ (OrdIso( E , ⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅)β€˜(2nd β€˜π‘)) = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’))
6463eqeq2d 2742 . . . . . 6 (𝑐 = βŸ¨π‘‘, π‘’βŸ© β†’ (𝑏 = (OrdIso( E , ⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅)β€˜(2nd β€˜π‘)) ↔ 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)))
6564rexxp 5842 . . . . 5 (βˆƒπ‘ ∈ (𝐴 Γ— 𝐢)𝑏 = (OrdIso( E , ⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅)β€˜(2nd β€˜π‘)) ↔ βˆƒπ‘‘ ∈ 𝐴 βˆƒπ‘’ ∈ 𝐢 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’))
6654, 55, 653imtr4g 296 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (𝑏 ∈ βˆͺ π‘Ž ∈ 𝐴 𝐡 β†’ βˆƒπ‘ ∈ (𝐴 Γ— 𝐢)𝑏 = (OrdIso( E , ⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅)β€˜(2nd β€˜π‘))))
6766imp 406 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) ∧ 𝑏 ∈ βˆͺ π‘Ž ∈ 𝐴 𝐡) β†’ βˆƒπ‘ ∈ (𝐴 Γ— 𝐢)𝑏 = (OrdIso( E , ⦋(1st β€˜π‘) / π‘Žβ¦Œπ΅)β€˜(2nd β€˜π‘)))
688, 67wdomd 9580 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ βˆͺ π‘Ž ∈ 𝐴 𝐡 β‰Ό* (𝐴 Γ— 𝐢))
69 hsmexlem.g . . 3 𝐺 = OrdIso( E , βˆͺ π‘Ž ∈ 𝐴 𝐡)
7069hsmexlem1 10425 . 2 ((βˆͺ π‘Ž ∈ 𝐴 𝐡 βŠ† On ∧ βˆͺ π‘Ž ∈ 𝐴 𝐡 β‰Ό* (𝐴 Γ— 𝐢)) β†’ dom 𝐺 ∈ (harβ€˜π’« (𝐴 Γ— 𝐢)))
716, 68, 70syl2anc 583 1 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ dom 𝐺 ∈ (harβ€˜π’« (𝐴 Γ— 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  Vcvv 3473  β¦‹csb 3893   βŠ† wss 3948  π’« cpw 4602  βŸ¨cop 4634  βˆͺ ciun 4997   class class class wbr 5148   E cep 5579   Γ— cxp 5674  dom cdm 5676  ran crn 5677  Oncon0 6364  βŸΆwf 6539  β€“ontoβ†’wfo 6541  β€˜cfv 6543  1st c1st 7977  2nd c2nd 7978  Smo wsmo 8349  OrdIsocoi 9508  harchar 9555   β‰Ό* cwdom 9563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-smo 8350  df-recs 8375  df-en 8944  df-dom 8945  df-sdom 8946  df-oi 9509  df-har 9556  df-wdom 9564
This theorem is referenced by:  hsmexlem3  10427
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