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Theorem cnrest2 23010
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 22964 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top)
21a1i 11 . . 3 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐽 ∈ Top))
3 eqid 2730 . . . . . . . 8 βˆͺ 𝐽 = βˆͺ 𝐽
4 eqid 2730 . . . . . . . 8 βˆͺ 𝐾 = βˆͺ 𝐾
53, 4cnf 22970 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹:βˆͺ 𝐽⟢βˆͺ 𝐾)
65ffnd 6717 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) β†’ 𝐹 Fn βˆͺ 𝐽)
76a1i 11 . . . . 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 1135 . . . . 5 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ ran 𝐹 βŠ† 𝐡)
97, 8jctird 525 . . . 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 6546 . . . 4 (𝐹:βˆͺ 𝐽⟢𝐡 ↔ (𝐹 Fn βˆͺ 𝐽 ∧ ran 𝐹 βŠ† 𝐡))
119, 10imbitrrdi 251 . . 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 511 . 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 22964 . . . . 5 (𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡)) β†’ 𝐽 ∈ Top)
1413adantl 480 . . . 4 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ 𝐹 ∈ (𝐽 Cn (𝐾 β†Ύt 𝐡))) β†’ 𝐽 ∈ Top)
15 toptopon2 22640 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOnβ€˜βˆͺ 𝐽))
1614, 15sylib 217 . . . . 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 22885 . . . . . . 7 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐾 β†Ύt 𝐡) ∈ (TopOnβ€˜π΅))
18173adant2 1129 . . . . . 6 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) β†’ (𝐾 β†Ύt 𝐡) ∈ (TopOnβ€˜π΅))
1918adantr 479 . . . . 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 483 . . . . 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 22973 . . . . 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 1369 . . . 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 510 . . 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 411 . 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 3476 . . . . . . . . 9 π‘₯ ∈ V
2625inex1 5316 . . . . . . . 8 (π‘₯ ∩ 𝐡) ∈ V
2726a1i 11 . . . . . . 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 1189 . . . . . . . 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 22648 . . . . . . . . . 10 (𝐾 ∈ (TopOnβ€˜π‘Œ) β†’ π‘Œ ∈ 𝐾)
3028, 29syl 17 . . . . . . . . 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 1191 . . . . . . . . 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 5323 . . . . . . . 8 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐡 ∈ V)
33 elrest 17377 . . . . . . . 8 ((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ V) β†’ (𝑦 ∈ (𝐾 β†Ύt 𝐡) ↔ βˆƒπ‘₯ ∈ 𝐾 𝑦 = (π‘₯ ∩ 𝐡)))
3428, 32, 33syl2anc 582 . . . . . . 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 6054 . . . . . . . . 9 (𝑦 = (π‘₯ ∩ 𝐡) β†’ (◑𝐹 β€œ 𝑦) = (◑𝐹 β€œ (π‘₯ ∩ 𝐡)))
3635eleq1d 2816 . . . . . . . 8 (𝑦 = (π‘₯ ∩ 𝐡) β†’ ((◑𝐹 β€œ 𝑦) ∈ 𝐽 ↔ (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) ∈ 𝐽))
3736adantl 480 . . . . . . 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 5407 . . . . . 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 774 . . . . . . . . . 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 6719 . . . . . . . . . 10 (𝐹:βˆͺ 𝐽⟢𝐡 β†’ Fun 𝐹)
41 inpreima 7064 . . . . . . . . . 10 (Fun 𝐹 β†’ (◑𝐹 β€œ (π‘₯ ∩ 𝐡)) = ((◑𝐹 β€œ π‘₯) ∩ (◑𝐹 β€œ 𝐡)))
4239, 40, 413syl 18 . . . . . . . . 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 6079 . . . . . . . . . . . 12 (◑𝐹 β€œ π‘₯) βŠ† dom 𝐹
44 cnvimarndm 6080 . . . . . . . . . . . 12 (◑𝐹 β€œ ran 𝐹) = dom 𝐹
4543, 44sseqtrri 4018 . . . . . . . . . . 11 (◑𝐹 β€œ π‘₯) βŠ† (◑𝐹 β€œ ran 𝐹)
46 simpll2 1211 . . . . . . . . . . . 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 6100 . . . . . . . . . . . 12 (ran 𝐹 βŠ† 𝐡 β†’ (◑𝐹 β€œ ran 𝐹) βŠ† (◑𝐹 β€œ 𝐡))
4846, 47syl 17 . . . . . . . . . . 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 3992 . . . . . . . . . 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 3964 . . . . . . . . . 10 ((◑𝐹 β€œ π‘₯) βŠ† (◑𝐹 β€œ 𝐡) ↔ ((◑𝐹 β€œ π‘₯) ∩ (◑𝐹 β€œ 𝐡)) = (◑𝐹 β€œ π‘₯))
5149, 50sylib 217 . . . . . . . . 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 2770 . . . . . . . 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 2816 . . . . . . 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 3173 . . . . . 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 769 . . . . . . . 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 6734 . . . . . . 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 531 . . . . . 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 305 . . . . 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 531 . . . . 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 278 . . . 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 767 . . . . . 6 (((𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† π‘Œ) ∧ (𝐽 ∈ Top ∧ 𝐹:βˆͺ 𝐽⟢𝐡)) β†’ 𝐽 ∈ Top)
6261, 15sylib 217 . . . . 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 22959 . . . . 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 582 . . . 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 479 . . . . 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 22959 . . . . 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 582 . . . 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 310 . . 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 411 . 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 378 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 setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  βˆͺ cuni 4907  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β€œ cima 5678  Fun wfun 6536   Fn wfn 6537  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411   β†Ύt crest 17370  Topctop 22615  TopOnctopon 22632   Cn ccn 22948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-map 8824  df-en 8942  df-fin 8945  df-fi 9408  df-rest 17372  df-topgen 17393  df-top 22616  df-topon 22633  df-bases 22669  df-cn 22951
This theorem is referenced by:  cnrest2r  23011  rncmp  23120  connima  23149  conncn  23150  kgencn2  23281  kgencn3  23282  qtoprest  23441  hmeores  23495  efmndtmd  23825  submtmd  23828  subgtgp  23829  symgtgp  23830  metdcn2  24575  metdscn2  24593  cnmptre  24668  iimulcn  24681  iimulcnOLD  24682  icchmeo  24685  icchmeoOLD  24686  evth  24705  evth2  24706  lebnumlem2  24708  reparphti  24743  reparphtiOLD  24744  efrlim  26710  rmulccn  33206  raddcn  33207  xrge0mulc1cn  33219  cvxpconn  34531  cvxsconn  34532  cvmliftmolem1  34570  cvmliftlem8  34581  cvmlift2lem9  34600  cvmlift3lem6  34613  gg-rmulccn  35465  ivthALT  35523  knoppcnlem10  35681  broucube  36825  areacirclem2  36880  cnres2  36934  cnresima  36935  refsumcn  44016  icccncfext  44901
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