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Theorem cnrest2 13706
Description: Equivalence of continuity in the parent topology and continuity in a subspace. (Contributed by Jeff Hankins, 10-Jul-2009.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
cnrest2 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))))
Proof of Theorem cnrest2
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cntop1 13671 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top)
21a1i 9 . . 3 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top))
3 eqid 2177 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
4 eqid 2177 . . . . . . . 8 βˆͺ 𝐾 = βˆͺ 𝐾
53, 4cnf 13674 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
65ffnd 5366 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹 Fn βˆͺ 𝐽)
76a1i 9 . . . . 5 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹 Fn βˆͺ 𝐽))
8 simp2 998 . . . . 5 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ ran 𝐹 βŠ† 𝐡)
97, 8jctird 317 . . . 4 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ (𝐹 Fn βˆͺ 𝐽 ∧ ran 𝐹 βŠ† 𝐡)))
10 df-f 5220 . . . 4 (𝐹:βˆͺ 𝐽⟢𝐡 ↔ (𝐹 Fn βˆͺ 𝐽 ∧ ran 𝐹 βŠ† 𝐡))
119, 10imbitrrdi 162 . . 3 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:βˆͺ 𝐽⟢𝐡))
122, 11jcad 307 . 2 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)))
13 cntop1 13671 . . . . 5 (𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡)) β†’ 𝐽 ∈ Top)
1413adantl 277 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))) β†’ 𝐽 ∈ Top)
153toptopon 13488 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
1614, 15sylib 122 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
17 resttopon 13641 . . . . . . 7 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐾 β†Ύt 𝐡) ∈ (TopOnβ€˜π΅))
18173adant2 1016 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐾 β†Ύt 𝐡) ∈ (TopOnβ€˜π΅))
1918adantr 276 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))) β†’ (𝐾 β†Ύt 𝐡) ∈ (TopOnβ€˜π΅))
20 simpr 110 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))) β†’ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡)))
21 cnf2 13675 . . . . 5 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ (𝐾 β†Ύt 𝐡) ∈ (TopOnβ€˜π΅) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))) β†’ 𝐹:βˆͺ 𝐽⟢𝐡)
2216, 19, 20, 21syl3anc 1238 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))) β†’ 𝐹:βˆͺ 𝐽⟢𝐡)
2314, 22jca 306 . . 3 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))) β†’ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡))
2423ex 115 . 2 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡)) β†’ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)))
25 vex 2740 . . . . . . . . 9 π‘₯ ∈ V
2625inex1 4137 . . . . . . . 8 (π‘₯ ∩ 𝐡) ∈ V
2726a1i 9 . . . . . . 7 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ (π‘₯ ∩ 𝐡) ∈ V)
28 simpl1 1000 . . . . . . . 8 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
29 toponmax 13495 . . . . . . . . . 10 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐾)
3028, 29syl 14 . . . . . . . . 9 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ π‘Œ ∈ 𝐾)
31 simpl3 1002 . . . . . . . . 9 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐡 βŠ† π‘Œ)
3230, 31ssexd 4143 . . . . . . . 8 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐡 ∈ V)
33 elrest 12694 . . . . . . . 8 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ V) β†’ (𝑦 ∈ (𝐾 β†Ύt 𝐡) ↔ βˆƒπ‘₯ ∈ 𝐾 𝑦 = (π‘₯ ∩ 𝐡)))
3428, 32, 33syl2anc 411 . . . . . . 7 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (𝑦 ∈ (𝐾 β†Ύt 𝐡) ↔ βˆƒπ‘₯ ∈ 𝐾 𝑦 = (π‘₯ ∩ 𝐡)))
35 imaeq2 4966 . . . . . . . . 9 (𝑦 = (π‘₯ ∩ 𝐡) β†’ (◑𝐹 β€œ 𝑦) = (◑𝐹 β€œ (π‘₯ ∩ 𝐡)))
3635eleq1d 2246 . . . . . . . 8 (𝑦 = (π‘₯ ∩ 𝐡) β†’ ((◑𝐹 β€œ 𝑦) ∈ 𝐽 ↔ (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) ∈ 𝐽))
3736adantl 277 . . . . . . 7 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ 𝑦 = (π‘₯ ∩ 𝐡)) β†’ ((◑𝐹 β€œ 𝑦) ∈ 𝐽 ↔ (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) ∈ 𝐽))
3827, 34, 37ralxfr2d 4464 . . . . . 6 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (βˆ€π‘¦ ∈ (𝐾 β†Ύt 𝐡)(◑𝐹 β€œ 𝑦) ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) ∈ 𝐽))
39 simplrr 536 . . . . . . . . . 10 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ 𝐹:βˆͺ 𝐽⟢𝐡)
40 ffun 5368 . . . . . . . . . 10 (𝐹:βˆͺ 𝐽⟢𝐡 β†’ Fun 𝐹)
41 inpreima 5642 . . . . . . . . . 10 (Fun 𝐹 β†’ (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) = ((◑𝐹 β€œ π‘₯) ∩ (◑𝐹 β€œ 𝐡)))
4239, 40, 413syl 17 . . . . . . . . 9 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) = ((◑𝐹 β€œ π‘₯) ∩ (◑𝐹 β€œ 𝐡)))
43 cnvimass 4991 . . . . . . . . . . . 12 (◑𝐹 β€œ π‘₯) βŠ† dom 𝐹
44 cnvimarndm 4992 . . . . . . . . . . . 12 (◑𝐹 β€œ ran 𝐹) = dom 𝐹
4543, 44sseqtrri 3190 . . . . . . . . . . 11 (◑𝐹 β€œ π‘₯) βŠ† (◑𝐹 β€œ ran 𝐹)
46 simpll2 1037 . . . . . . . . . . . 12 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ ran 𝐹 βŠ† 𝐡)
47 imass2 5004 . . . . . . . . . . . 12 (ran 𝐹 βŠ† 𝐡 β†’ (◑𝐹 β€œ ran 𝐹) βŠ† (◑𝐹 β€œ 𝐡))
4846, 47syl 14 . . . . . . . . . . 11 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ ran 𝐹) βŠ† (◑𝐹 β€œ 𝐡))
4945, 48sstrid 3166 . . . . . . . . . 10 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ π‘₯) βŠ† (◑𝐹 β€œ 𝐡))
50 df-ss 3142 . . . . . . . . . 10 ((◑𝐹 β€œ π‘₯) βŠ† (◑𝐹 β€œ 𝐡) ↔ ((◑𝐹 β€œ π‘₯) ∩ (◑𝐹 β€œ 𝐡)) = (◑𝐹 β€œ π‘₯))
5149, 50sylib 122 . . . . . . . . 9 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ ((◑𝐹 β€œ π‘₯) ∩ (◑𝐹 β€œ 𝐡)) = (◑𝐹 β€œ π‘₯))
5242, 51eqtrd 2210 . . . . . . . 8 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) = (◑𝐹 β€œ π‘₯))
5352eleq1d 2246 . . . . . . 7 ((((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) ∧ π‘₯ ∈ 𝐾) β†’ ((◑𝐹 β€œ (π‘₯ ∩ 𝐡)) ∈ 𝐽 ↔ (◑𝐹 β€œ π‘₯) ∈ 𝐽))
5453ralbidva 2473 . . . . . 6 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) ∈ 𝐽 ↔ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽))
55 simprr 531 . . . . . . . 8 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐹:βˆͺ 𝐽⟢𝐡)
5655, 31fssd 5378 . . . . . . 7 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐹:βˆͺ π½βŸΆπ‘Œ)
5756biantrurd 305 . . . . . 6 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽 ↔ (𝐹:βˆͺ π½βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
5838, 54, 573bitrrd 215 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ ((𝐹:βˆͺ π½βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽) ↔ βˆ€π‘¦ ∈ (𝐾 β†Ύt 𝐡)(◑𝐹 β€œ 𝑦) ∈ 𝐽))
5955biantrurd 305 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (βˆ€π‘¦ ∈ (𝐾 β†Ύt 𝐡)(◑𝐹 β€œ 𝑦) ∈ 𝐽 ↔ (𝐹:βˆͺ 𝐽⟢𝐡 ∧ βˆ€π‘¦ ∈ (𝐾 β†Ύt 𝐡)(◑𝐹 β€œ 𝑦) ∈ 𝐽)))
6058, 59bitrd 188 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ ((𝐹:βˆͺ π½βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽) ↔ (𝐹:βˆͺ 𝐽⟢𝐡 ∧ βˆ€π‘¦ ∈ (𝐾 β†Ύt 𝐡)(◑𝐹 β€œ 𝑦) ∈ 𝐽)))
61 simprl 529 . . . . . 6 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐽 ∈ Top)
6261, 15sylib 122 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
63 iscn 13667 . . . . 5 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:βˆͺ π½βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
6462, 28, 63syl2anc 411 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:βˆͺ π½βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝐾 (◑𝐹 β€œ π‘₯) ∈ 𝐽)))
6518adantr 276 . . . . 5 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (𝐾 β†Ύt 𝐡) ∈ (TopOnβ€˜π΅))
66 iscn 13667 . . . . 5 ((𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽) ∧ (𝐾 β†Ύt 𝐡) ∈ (TopOnβ€˜π΅)) β†’ (𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡)) ↔ (𝐹:βˆͺ 𝐽⟢𝐡 ∧ βˆ€π‘¦ ∈ (𝐾 β†Ύt 𝐡)(◑𝐹 β€œ 𝑦) ∈ 𝐽)))
6762, 65, 66syl2anc 411 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡)) ↔ (𝐹:βˆͺ 𝐽⟢𝐡 ∧ βˆ€π‘¦ ∈ (𝐾 β†Ύt 𝐡)(◑𝐹 β€œ 𝑦) ∈ 𝐽)))
6860, 64, 673bitr4d 220 . . 3 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))))
6968ex 115 . 2 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ ((𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡)))))
7012, 24, 69pm5.21ndd 705 1 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  βˆ€wral 2455  βˆƒwrex 2456  Vcvv 2737   ∩ cin 3128   βŠ† wss 3129  βˆͺ cuni 3809  β—‘ccnv 4625  dom cdm 4626  ran crn 4627   β€œ cima 4629  Fun wfun 5210   Fn wfn 5211  βŸΆwf 5212  β€˜cfv 5216  (class class class)co 5874   β†Ύt crest 12687  Topctop 13467  TopOnctopon 13480   Cn ccn 13655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-map 6649  df-rest 12689  df-topgen 12708  df-top 13468  df-topon 13481  df-bases 13513  df-cn 13658
This theorem is referenced by:  cnrest2r  13707  hmeores  13785
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