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Theorem cnrest2 12186
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 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))))

Proof of Theorem cnrest2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cntop1 12151 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
21a1i 9 . . 3 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top))
3 eqid 2100 . . . . . . . 8 𝐽 = 𝐽
4 eqid 2100 . . . . . . . 8 𝐾 = 𝐾
53, 4cnf 12154 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
65ffnd 5209 . . . . . 6 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 Fn 𝐽)
76a1i 9 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹 Fn 𝐽))
8 simp2 950 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → ran 𝐹𝐵)
97, 8jctird 313 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 Fn 𝐽 ∧ ran 𝐹𝐵)))
10 df-f 5063 . . . 4 (𝐹: 𝐽𝐵 ↔ (𝐹 Fn 𝐽 ∧ ran 𝐹𝐵))
119, 10syl6ibr 161 . . 3 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝐵))
122, 11jcad 303 . 2 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)))
13 cntop1 12151 . . . . 5 (𝐹 ∈ (𝐽 Cn (𝐾t 𝐵)) → 𝐽 ∈ Top)
1413adantl 273 . . . 4 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → 𝐽 ∈ Top)
153toptopon 11967 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
1614, 15sylib 121 . . . . 5 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → 𝐽 ∈ (TopOn‘ 𝐽))
17 resttopon 12122 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
18173adant2 968 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
1918adantr 272 . . . . 5 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
20 simpr 109 . . . . 5 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵)))
21 cnf2 12155 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝐾t 𝐵) ∈ (TopOn‘𝐵) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → 𝐹: 𝐽𝐵)
2216, 19, 20, 21syl3anc 1184 . . . 4 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → 𝐹: 𝐽𝐵)
2314, 22jca 302 . . 3 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵))
2423ex 114 . 2 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn (𝐾t 𝐵)) → (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)))
25 vex 2644 . . . . . . . . 9 𝑥 ∈ V
2625inex1 4002 . . . . . . . 8 (𝑥𝐵) ∈ V
2726a1i 9 . . . . . . 7 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → (𝑥𝐵) ∈ V)
28 simpl1 952 . . . . . . . 8 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐾 ∈ (TopOn‘𝑌))
29 toponmax 11974 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
3028, 29syl 14 . . . . . . . . 9 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝑌𝐾)
31 simpl3 954 . . . . . . . . 9 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐵𝑌)
3230, 31ssexd 4008 . . . . . . . 8 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐵 ∈ V)
33 elrest 11909 . . . . . . . 8 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ V) → (𝑦 ∈ (𝐾t 𝐵) ↔ ∃𝑥𝐾 𝑦 = (𝑥𝐵)))
3428, 32, 33syl2anc 406 . . . . . . 7 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (𝑦 ∈ (𝐾t 𝐵) ↔ ∃𝑥𝐾 𝑦 = (𝑥𝐵)))
35 imaeq2 4813 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → (𝐹𝑦) = (𝐹 “ (𝑥𝐵)))
3635eleq1d 2168 . . . . . . . 8 (𝑦 = (𝑥𝐵) → ((𝐹𝑦) ∈ 𝐽 ↔ (𝐹 “ (𝑥𝐵)) ∈ 𝐽))
3736adantl 273 . . . . . . 7 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑦 = (𝑥𝐵)) → ((𝐹𝑦) ∈ 𝐽 ↔ (𝐹 “ (𝑥𝐵)) ∈ 𝐽))
3827, 34, 37ralxfr2d 4323 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽 ↔ ∀𝑥𝐾 (𝐹 “ (𝑥𝐵)) ∈ 𝐽))
39 simplrr 506 . . . . . . . . . 10 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → 𝐹: 𝐽𝐵)
40 ffun 5211 . . . . . . . . . 10 (𝐹: 𝐽𝐵 → Fun 𝐹)
41 inpreima 5478 . . . . . . . . . 10 (Fun 𝐹 → (𝐹 “ (𝑥𝐵)) = ((𝐹𝑥) ∩ (𝐹𝐵)))
4239, 40, 413syl 17 . . . . . . . . 9 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → (𝐹 “ (𝑥𝐵)) = ((𝐹𝑥) ∩ (𝐹𝐵)))
43 cnvimass 4838 . . . . . . . . . . . 12 (𝐹𝑥) ⊆ dom 𝐹
44 cnvimarndm 4839 . . . . . . . . . . . 12 (𝐹 “ ran 𝐹) = dom 𝐹
4543, 44sseqtr4i 3082 . . . . . . . . . . 11 (𝐹𝑥) ⊆ (𝐹 “ ran 𝐹)
46 simpll2 989 . . . . . . . . . . . 12 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → ran 𝐹𝐵)
47 imass2 4851 . . . . . . . . . . . 12 (ran 𝐹𝐵 → (𝐹 “ ran 𝐹) ⊆ (𝐹𝐵))
4846, 47syl 14 . . . . . . . . . . 11 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → (𝐹 “ ran 𝐹) ⊆ (𝐹𝐵))
4945, 48syl5ss 3058 . . . . . . . . . 10 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ (𝐹𝐵))
50 df-ss 3034 . . . . . . . . . 10 ((𝐹𝑥) ⊆ (𝐹𝐵) ↔ ((𝐹𝑥) ∩ (𝐹𝐵)) = (𝐹𝑥))
5149, 50sylib 121 . . . . . . . . 9 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → ((𝐹𝑥) ∩ (𝐹𝐵)) = (𝐹𝑥))
5242, 51eqtrd 2132 . . . . . . . 8 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → (𝐹 “ (𝑥𝐵)) = (𝐹𝑥))
5352eleq1d 2168 . . . . . . 7 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → ((𝐹 “ (𝑥𝐵)) ∈ 𝐽 ↔ (𝐹𝑥) ∈ 𝐽))
5453ralbidva 2392 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (∀𝑥𝐾 (𝐹 “ (𝑥𝐵)) ∈ 𝐽 ↔ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽))
55 simprr 502 . . . . . . . 8 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐹: 𝐽𝐵)
5655, 31fssd 5221 . . . . . . 7 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐹: 𝐽𝑌)
5756biantrurd 301 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽 ↔ (𝐹: 𝐽𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
5838, 54, 573bitrrd 214 . . . . 5 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → ((𝐹: 𝐽𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽) ↔ ∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽))
5955biantrurd 301 . . . . 5 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽 ↔ (𝐹: 𝐽𝐵 ∧ ∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽)))
6058, 59bitrd 187 . . . 4 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → ((𝐹: 𝐽𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽) ↔ (𝐹: 𝐽𝐵 ∧ ∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽)))
61 simprl 501 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐽 ∈ Top)
6261, 15sylib 121 . . . . 5 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐽 ∈ (TopOn‘ 𝐽))
63 iscn 12147 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹: 𝐽𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
6462, 28, 63syl2anc 406 . . . 4 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹: 𝐽𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
6518adantr 272 . . . . 5 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
66 iscn 12147 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝐾t 𝐵) ∈ (TopOn‘𝐵)) → (𝐹 ∈ (𝐽 Cn (𝐾t 𝐵)) ↔ (𝐹: 𝐽𝐵 ∧ ∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽)))
6762, 65, 66syl2anc 406 . . . 4 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (𝐹 ∈ (𝐽 Cn (𝐾t 𝐵)) ↔ (𝐹: 𝐽𝐵 ∧ ∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽)))
6860, 64, 673bitr4d 219 . . 3 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))))
6968ex 114 . 2 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → ((𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵)))))
7012, 24, 69pm5.21ndd 662 1 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 930   = wceq 1299  wcel 1448  wral 2375  wrex 2376  Vcvv 2641  cin 3020  wss 3021   cuni 3683  ccnv 4476  dom cdm 4477  ran crn 4478  cima 4480  Fun wfun 5053   Fn wfn 5054  wf 5055  cfv 5059  (class class class)co 5706  t crest 11902  Topctop 11946  TopOnctopon 11959   Cn ccn 12136
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-map 6474  df-rest 11904  df-topgen 11923  df-top 11947  df-topon 11960  df-bases 11992  df-cn 12139
This theorem is referenced by:  cnrest2r  12187
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