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Theorem tgrest 22662
Description: A subspace can be generated by restricted sets from a basis for the original topology. (Contributed by Mario Carneiro, 19-Mar-2015.) (Proof shortened by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
tgrest ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (topGenβ€˜(𝐡 β†Ύt 𝐴)) = ((topGenβ€˜π΅) β†Ύt 𝐴))
Proof of Theorem tgrest
Dummy variables 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 7441 . . . . 5 (𝐡 β†Ύt 𝐴) ∈ V
2 eltg3 22464 . . . . 5 ((𝐡 β†Ύt 𝐴) ∈ V β†’ (π‘₯ ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)) ↔ βˆƒπ‘¦(𝑦 βŠ† (𝐡 β†Ύt 𝐴) ∧ π‘₯ = βˆͺ 𝑦)))
31, 2ax-mp 5 . . . 4 (π‘₯ ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)) ↔ βˆƒπ‘¦(𝑦 βŠ† (𝐡 β†Ύt 𝐴) ∧ π‘₯ = βˆͺ 𝑦))
4 simpll 765 . . . . . . . . 9 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑦 βŠ† (𝐡 β†Ύt 𝐴)) β†’ 𝐡 ∈ 𝑉)
5 funmpt 6586 . . . . . . . . . 10 Fun (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴))
65a1i 11 . . . . . . . . 9 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑦 βŠ† (𝐡 β†Ύt 𝐴)) β†’ Fun (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)))
7 restval 17371 . . . . . . . . . . . 12 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (𝐡 β†Ύt 𝐴) = ran (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)))
87sseq2d 4014 . . . . . . . . . . 11 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (𝑦 βŠ† (𝐡 β†Ύt 𝐴) ↔ 𝑦 βŠ† ran (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴))))
98biimpa 477 . . . . . . . . . 10 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑦 βŠ† (𝐡 β†Ύt 𝐴)) β†’ 𝑦 βŠ† ran (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)))
10 vex 3478 . . . . . . . . . . . . 13 π‘₯ ∈ V
1110inex1 5317 . . . . . . . . . . . 12 (π‘₯ ∩ 𝐴) ∈ V
1211rgenw 3065 . . . . . . . . . . 11 βˆ€π‘₯ ∈ 𝐡 (π‘₯ ∩ 𝐴) ∈ V
13 eqid 2732 . . . . . . . . . . . 12 (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) = (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴))
1413fnmpt 6690 . . . . . . . . . . 11 (βˆ€π‘₯ ∈ 𝐡 (π‘₯ ∩ 𝐴) ∈ V β†’ (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) Fn 𝐡)
15 fnima 6680 . . . . . . . . . . 11 ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) Fn 𝐡 β†’ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝐡) = ran (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)))
1612, 14, 15mp2b 10 . . . . . . . . . 10 ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝐡) = ran (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴))
179, 16sseqtrrdi 4033 . . . . . . . . 9 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑦 βŠ† (𝐡 β†Ύt 𝐴)) β†’ 𝑦 βŠ† ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝐡))
18 ssimaexg 6977 . . . . . . . . 9 ((𝐡 ∈ 𝑉 ∧ Fun (π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) ∧ 𝑦 βŠ† ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝐡)) β†’ βˆƒπ‘§(𝑧 βŠ† 𝐡 ∧ 𝑦 = ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧)))
194, 6, 17, 18syl3anc 1371 . . . . . . . 8 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑦 βŠ† (𝐡 β†Ύt 𝐴)) β†’ βˆƒπ‘§(𝑧 βŠ† 𝐡 ∧ 𝑦 = ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧)))
20 df-ima 5689 . . . . . . . . . . . . . . . . 17 ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) = ran ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β†Ύ 𝑧)
21 resmpt 6037 . . . . . . . . . . . . . . . . . . 19 (𝑧 βŠ† 𝐡 β†’ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β†Ύ 𝑧) = (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)))
2221adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β†Ύ 𝑧) = (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)))
2322rneqd 5937 . . . . . . . . . . . . . . . . 17 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ ran ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β†Ύ 𝑧) = ran (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)))
2420, 23eqtrid 2784 . . . . . . . . . . . . . . . 16 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) = ran (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)))
2524unieqd 4922 . . . . . . . . . . . . . . 15 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) = βˆͺ ran (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)))
2611dfiun3 5965 . . . . . . . . . . . . . . 15 βˆͺ π‘₯ ∈ 𝑧 (π‘₯ ∩ 𝐴) = βˆͺ ran (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴))
2725, 26eqtr4di 2790 . . . . . . . . . . . . . 14 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) = βˆͺ π‘₯ ∈ 𝑧 (π‘₯ ∩ 𝐴))
28 iunin1 5075 . . . . . . . . . . . . . 14 βˆͺ π‘₯ ∈ 𝑧 (π‘₯ ∩ 𝐴) = (βˆͺ π‘₯ ∈ 𝑧 π‘₯ ∩ 𝐴)
2927, 28eqtrdi 2788 . . . . . . . . . . . . 13 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) = (βˆͺ π‘₯ ∈ 𝑧 π‘₯ ∩ 𝐴))
30 fvex 6904 . . . . . . . . . . . . . 14 (topGenβ€˜π΅) ∈ V
31 simpr 485 . . . . . . . . . . . . . 14 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ 𝐴 ∈ π‘Š)
32 uniiun 5061 . . . . . . . . . . . . . . . 16 βˆͺ 𝑧 = βˆͺ π‘₯ ∈ 𝑧 π‘₯
33 eltg3i 22463 . . . . . . . . . . . . . . . 16 ((𝐡 ∈ 𝑉 ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ 𝑧 ∈ (topGenβ€˜π΅))
3432, 33eqeltrrid 2838 . . . . . . . . . . . . . . 15 ((𝐡 ∈ 𝑉 ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ π‘₯ ∈ 𝑧 π‘₯ ∈ (topGenβ€˜π΅))
3534adantlr 713 . . . . . . . . . . . . . 14 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ π‘₯ ∈ 𝑧 π‘₯ ∈ (topGenβ€˜π΅))
36 elrestr 17373 . . . . . . . . . . . . . 14 (((topGenβ€˜π΅) ∈ V ∧ 𝐴 ∈ π‘Š ∧ βˆͺ π‘₯ ∈ 𝑧 π‘₯ ∈ (topGenβ€˜π΅)) β†’ (βˆͺ π‘₯ ∈ 𝑧 π‘₯ ∩ 𝐴) ∈ ((topGenβ€˜π΅) β†Ύt 𝐴))
3730, 31, 35, 36mp3an2ani 1468 . . . . . . . . . . . . 13 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ (βˆͺ π‘₯ ∈ 𝑧 π‘₯ ∩ 𝐴) ∈ ((topGenβ€˜π΅) β†Ύt 𝐴))
3829, 37eqeltrd 2833 . . . . . . . . . . . 12 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) ∈ ((topGenβ€˜π΅) β†Ύt 𝐴))
39 unieq 4919 . . . . . . . . . . . . 13 (𝑦 = ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) β†’ βˆͺ 𝑦 = βˆͺ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧))
4039eleq1d 2818 . . . . . . . . . . . 12 (𝑦 = ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) β†’ (βˆͺ 𝑦 ∈ ((topGenβ€˜π΅) β†Ύt 𝐴) ↔ βˆͺ ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
4138, 40syl5ibrcom 246 . . . . . . . . . . 11 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ (𝑦 = ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧) β†’ βˆͺ 𝑦 ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
4241expimpd 454 . . . . . . . . . 10 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ ((𝑧 βŠ† 𝐡 ∧ 𝑦 = ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧)) β†’ βˆͺ 𝑦 ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
4342exlimdv 1936 . . . . . . . . 9 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (βˆƒπ‘§(𝑧 βŠ† 𝐡 ∧ 𝑦 = ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧)) β†’ βˆͺ 𝑦 ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
4443adantr 481 . . . . . . . 8 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑦 βŠ† (𝐡 β†Ύt 𝐴)) β†’ (βˆƒπ‘§(𝑧 βŠ† 𝐡 ∧ 𝑦 = ((π‘₯ ∈ 𝐡 ↦ (π‘₯ ∩ 𝐴)) β€œ 𝑧)) β†’ βˆͺ 𝑦 ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
4519, 44mpd 15 . . . . . . 7 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑦 βŠ† (𝐡 β†Ύt 𝐴)) β†’ βˆͺ 𝑦 ∈ ((topGenβ€˜π΅) β†Ύt 𝐴))
46 eleq1 2821 . . . . . . 7 (π‘₯ = βˆͺ 𝑦 β†’ (π‘₯ ∈ ((topGenβ€˜π΅) β†Ύt 𝐴) ↔ βˆͺ 𝑦 ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
4745, 46syl5ibrcom 246 . . . . . 6 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑦 βŠ† (𝐡 β†Ύt 𝐴)) β†’ (π‘₯ = βˆͺ 𝑦 β†’ π‘₯ ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
4847expimpd 454 . . . . 5 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ ((𝑦 βŠ† (𝐡 β†Ύt 𝐴) ∧ π‘₯ = βˆͺ 𝑦) β†’ π‘₯ ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
4948exlimdv 1936 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (βˆƒπ‘¦(𝑦 βŠ† (𝐡 β†Ύt 𝐴) ∧ π‘₯ = βˆͺ 𝑦) β†’ π‘₯ ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
503, 49biimtrid 241 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (π‘₯ ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)) β†’ π‘₯ ∈ ((topGenβ€˜π΅) β†Ύt 𝐴)))
5150ssrdv 3988 . 2 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (topGenβ€˜(𝐡 β†Ύt 𝐴)) βŠ† ((topGenβ€˜π΅) β†Ύt 𝐴))
52 restval 17371 . . . 4 (((topGenβ€˜π΅) ∈ V ∧ 𝐴 ∈ π‘Š) β†’ ((topGenβ€˜π΅) β†Ύt 𝐴) = ran (𝑀 ∈ (topGenβ€˜π΅) ↦ (𝑀 ∩ 𝐴)))
5330, 31, 52sylancr 587 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ ((topGenβ€˜π΅) β†Ύt 𝐴) = ran (𝑀 ∈ (topGenβ€˜π΅) ↦ (𝑀 ∩ 𝐴)))
54 eltg3 22464 . . . . . . . 8 (𝐡 ∈ 𝑉 β†’ (𝑀 ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘§(𝑧 βŠ† 𝐡 ∧ 𝑀 = βˆͺ 𝑧)))
5554adantr 481 . . . . . . 7 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (𝑀 ∈ (topGenβ€˜π΅) ↔ βˆƒπ‘§(𝑧 βŠ† 𝐡 ∧ 𝑀 = βˆͺ 𝑧)))
5632ineq1i 4208 . . . . . . . . . . . 12 (βˆͺ 𝑧 ∩ 𝐴) = (βˆͺ π‘₯ ∈ 𝑧 π‘₯ ∩ 𝐴)
5756, 28eqtr4i 2763 . . . . . . . . . . 11 (βˆͺ 𝑧 ∩ 𝐴) = βˆͺ π‘₯ ∈ 𝑧 (π‘₯ ∩ 𝐴)
58 simplll 773 . . . . . . . . . . . . . . . 16 ((((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) ∧ π‘₯ ∈ 𝑧) β†’ 𝐡 ∈ 𝑉)
59 simpllr 774 . . . . . . . . . . . . . . . 16 ((((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) ∧ π‘₯ ∈ 𝑧) β†’ 𝐴 ∈ π‘Š)
60 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ 𝑧 βŠ† 𝐡)
6160sselda 3982 . . . . . . . . . . . . . . . 16 ((((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) ∧ π‘₯ ∈ 𝑧) β†’ π‘₯ ∈ 𝐡)
62 elrestr 17373 . . . . . . . . . . . . . . . 16 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š ∧ π‘₯ ∈ 𝐡) β†’ (π‘₯ ∩ 𝐴) ∈ (𝐡 β†Ύt 𝐴))
6358, 59, 61, 62syl3anc 1371 . . . . . . . . . . . . . . 15 ((((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) ∧ π‘₯ ∈ 𝑧) β†’ (π‘₯ ∩ 𝐴) ∈ (𝐡 β†Ύt 𝐴))
6463fmpttd 7114 . . . . . . . . . . . . . 14 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)):π‘§βŸΆ(𝐡 β†Ύt 𝐴))
6564frnd 6725 . . . . . . . . . . . . 13 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ ran (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)) βŠ† (𝐡 β†Ύt 𝐴))
66 eltg3i 22463 . . . . . . . . . . . . 13 (((𝐡 β†Ύt 𝐴) ∈ V ∧ ran (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)) βŠ† (𝐡 β†Ύt 𝐴)) β†’ βˆͺ ran (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)))
671, 65, 66sylancr 587 . . . . . . . . . . . 12 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ ran (π‘₯ ∈ 𝑧 ↦ (π‘₯ ∩ 𝐴)) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)))
6826, 67eqeltrid 2837 . . . . . . . . . . 11 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ βˆͺ π‘₯ ∈ 𝑧 (π‘₯ ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)))
6957, 68eqeltrid 2837 . . . . . . . . . 10 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ (βˆͺ 𝑧 ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)))
70 ineq1 4205 . . . . . . . . . . 11 (𝑀 = βˆͺ 𝑧 β†’ (𝑀 ∩ 𝐴) = (βˆͺ 𝑧 ∩ 𝐴))
7170eleq1d 2818 . . . . . . . . . 10 (𝑀 = βˆͺ 𝑧 β†’ ((𝑀 ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)) ↔ (βˆͺ 𝑧 ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴))))
7269, 71syl5ibrcom 246 . . . . . . . . 9 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑧 βŠ† 𝐡) β†’ (𝑀 = βˆͺ 𝑧 β†’ (𝑀 ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴))))
7372expimpd 454 . . . . . . . 8 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ ((𝑧 βŠ† 𝐡 ∧ 𝑀 = βˆͺ 𝑧) β†’ (𝑀 ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴))))
7473exlimdv 1936 . . . . . . 7 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (βˆƒπ‘§(𝑧 βŠ† 𝐡 ∧ 𝑀 = βˆͺ 𝑧) β†’ (𝑀 ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴))))
7555, 74sylbid 239 . . . . . 6 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (𝑀 ∈ (topGenβ€˜π΅) β†’ (𝑀 ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴))))
7675imp 407 . . . . 5 (((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) ∧ 𝑀 ∈ (topGenβ€˜π΅)) β†’ (𝑀 ∩ 𝐴) ∈ (topGenβ€˜(𝐡 β†Ύt 𝐴)))
7776fmpttd 7114 . . . 4 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (𝑀 ∈ (topGenβ€˜π΅) ↦ (𝑀 ∩ 𝐴)):(topGenβ€˜π΅)⟢(topGenβ€˜(𝐡 β†Ύt 𝐴)))
7877frnd 6725 . . 3 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ ran (𝑀 ∈ (topGenβ€˜π΅) ↦ (𝑀 ∩ 𝐴)) βŠ† (topGenβ€˜(𝐡 β†Ύt 𝐴)))
7953, 78eqsstrd 4020 . 2 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ ((topGenβ€˜π΅) β†Ύt 𝐴) βŠ† (topGenβ€˜(𝐡 β†Ύt 𝐴)))
8051, 79eqssd 3999 1 ((𝐡 ∈ 𝑉 ∧ 𝐴 ∈ π‘Š) β†’ (topGenβ€˜(𝐡 β†Ύt 𝐴)) = ((topGenβ€˜π΅) β†Ύt 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  βˆͺ cuni 4908  βˆͺ ciun 4997   ↦ cmpt 5231  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679  Fun wfun 6537   Fn wfn 6538  β€˜cfv 6543  (class class class)co 7408   β†Ύt crest 17365  topGenctg 17382
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 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-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-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-id 5574  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-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-rest 17367  df-topgen 17388
This theorem is referenced by:  resttop  22663  ordtrest2  22707  2ndcrest  22957  txrest  23134  xkoptsub  23157  xrtgioo  24321  ordtrest2NEW  32898  ptrest  36482
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