MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hsmexlem2 Structured version   Visualization version   GIF version

Theorem hsmexlem2 10424
Description: Lemma for hsmex 10429. Bound the order type of a union of sets of ordinals, each of limited order type. Vaguely reminiscent of unictb 10572 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 481 . . . . 5 ((𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) β†’ 𝐡 βŠ† On)
32ralimi 3083 . . . 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 1135 . 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 7739 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On) β†’ (𝐴 Γ— 𝐢) ∈ V)
873adant3 1132 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (𝐴 Γ— 𝐢) ∈ V)
9 nfv 1917 . . . . . . . . 9 β„²π‘Ž 𝐢 ∈ On
10 nfra1 3281 . . . . . . . . 9 β„²π‘Žβˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)
119, 10nfan 1902 . . . . . . . 8 β„²π‘Ž(𝐢 ∈ On ∧ βˆ€π‘Ž ∈ 𝐴 (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢))
12 rsp 3244 . . . . . . . . 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 407 . . . . . . . . . . . . 13 ((𝐢 ∈ On ∧ dom 𝐹 ∈ 𝐢) β†’ dom 𝐹 βŠ† 𝐢)
1514adantrl 714 . . . . . . . . . . . 12 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ dom 𝐹 βŠ† 𝐢)
16153adant3 1132 . . . . . . . . . . 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 9537 . . . . . . . . . . . . . . . . . 18 (𝐡 βŠ† On β†’ (Smo 𝐹 ∧ ran 𝐹 = 𝐡))
191, 18syl 17 . . . . . . . . . . . . . . . . 17 (𝐡 ∈ 𝒫 On β†’ (Smo 𝐹 ∧ ran 𝐹 = 𝐡))
2019ad2antrl 726 . . . . . . . . . . . . . . . 16 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ (Smo 𝐹 ∧ ran 𝐹 = 𝐡))
2120simprd 496 . . . . . . . . . . . . . . 15 ((𝐢 ∈ On ∧ (𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢)) β†’ ran 𝐹 = 𝐡)
2217oif 9527 . . . . . . . . . . . . . . 15 𝐹:dom 𝐹⟢𝐡
2321, 22jctil 520 . . . . . . . . . . . . . 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 497 . . . . . . . . . . . . 13 (𝐹:dom 𝐹–onto→𝐡 β†’ βˆ€π‘ ∈ 𝐡 βˆƒπ‘’ ∈ dom 𝐹 𝑏 = (πΉβ€˜π‘’))
28 rsp 3244 . . . . . . . . . . . . 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 1117 . . . . . . . . . . 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 1119 . . . . . . . . 9 (𝐢 ∈ On β†’ ((𝐡 ∈ 𝒫 On ∧ dom 𝐹 ∈ 𝐢) β†’ (𝑏 ∈ 𝐡 β†’ βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’))))
3412, 33sylan9r 509 . . . . . . . 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 3258 . . . . . . 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 1130 . . . . . 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 1917 . . . . . . 7 β„²π‘‘βˆƒπ‘’ ∈ 𝐢 𝑏 = (πΉβ€˜π‘’)
38 nfcv 2903 . . . . . . . 8 β„²π‘ŽπΆ
39 nfcv 2903 . . . . . . . . . . 11 β„²π‘Ž E
40 nfcsb1v 3918 . . . . . . . . . . 11 β„²π‘Žβ¦‹π‘‘ / π‘Žβ¦Œπ΅
4139, 40nfoi 9511 . . . . . . . . . 10 β„²π‘ŽOrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)
42 nfcv 2903 . . . . . . . . . 10 β„²π‘Žπ‘’
4341, 42nffv 6901 . . . . . . . . 9 β„²π‘Ž(OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)
4443nfeq2 2920 . . . . . . . 8 β„²π‘Ž 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)
4538, 44nfrexw 3310 . . . . . . 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 9510 . . . . . . . . . . . 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 2784 . . . . . . . . . 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 2743 . . . . . . . 8 (π‘Ž = 𝑑 β†’ (𝑏 = (πΉβ€˜π‘’) ↔ 𝑏 = (OrdIso( E , ⦋𝑑 / π‘Žβ¦Œπ΅)β€˜π‘’)))
5251rexbidv 3178 . . . . . . 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 3304 . . . . . 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 3478 . . . . . . . . . . 11 𝑑 ∈ V
57 vex 3478 . . . . . . . . . . 11 𝑒 ∈ V
5856, 57op1std 7987 . . . . . . . . . 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 9510 . . . . . . . . 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 7988 . . . . . . . 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 2743 . . . . . 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 295 . . . 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 407 . . 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 9578 . 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 10423 . 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 584 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474  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 7975  2nd c2nd 7976  Smo wsmo 8347  OrdIsocoi 9506  harchar 9553   β‰Ό* cwdom 9561
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 7367  df-ov 7414  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-smo 8348  df-recs 8373  df-en 8942  df-dom 8943  df-sdom 8944  df-oi 9507  df-har 9554  df-wdom 9562
This theorem is referenced by:  hsmexlem3  10425
  Copyright terms: Public domain W3C validator