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Theorem cnrest2 12442
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 12407 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
21a1i 9 . . 3 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top))
3 eqid 2140 . . . . . . . 8 𝐽 = 𝐽
4 eqid 2140 . . . . . . . 8 𝐾 = 𝐾
53, 4cnf 12410 . . . . . . 7 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽 𝐾)
65ffnd 5280 . . . . . 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 983 . . . . 5 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → ran 𝐹𝐵)
97, 8jctird 315 . . . 4 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐹 Fn 𝐽 ∧ ran 𝐹𝐵)))
10 df-f 5134 . . . 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 305 . 2 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)))
13 cntop1 12407 . . . . 5 (𝐹 ∈ (𝐽 Cn (𝐾t 𝐵)) → 𝐽 ∈ Top)
1413adantl 275 . . . 4 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → 𝐽 ∈ Top)
153toptopon 12222 . . . . . 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 12377 . . . . . . 7 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
18173adant2 1001 . . . . . 6 ((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
1918adantr 274 . . . . 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 12411 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝐾t 𝐵) ∈ (TopOn‘𝐵) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → 𝐹: 𝐽𝐵)
2216, 19, 20, 21syl3anc 1217 . . . 4 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ 𝐹 ∈ (𝐽 Cn (𝐾t 𝐵))) → 𝐹: 𝐽𝐵)
2314, 22jca 304 . . 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 2692 . . . . . . . . 9 𝑥 ∈ V
2625inex1 4069 . . . . . . . 8 (𝑥𝐵) ∈ V
2726a1i 9 . . . . . . 7 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → (𝑥𝐵) ∈ V)
28 simpl1 985 . . . . . . . 8 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐾 ∈ (TopOn‘𝑌))
29 toponmax 12229 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌𝐾)
3028, 29syl 14 . . . . . . . . 9 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝑌𝐾)
31 simpl3 987 . . . . . . . . 9 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐵𝑌)
3230, 31ssexd 4075 . . . . . . . 8 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐵 ∈ V)
33 elrest 12164 . . . . . . . 8 ((𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ V) → (𝑦 ∈ (𝐾t 𝐵) ↔ ∃𝑥𝐾 𝑦 = (𝑥𝐵)))
3428, 32, 33syl2anc 409 . . . . . . 7 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (𝑦 ∈ (𝐾t 𝐵) ↔ ∃𝑥𝐾 𝑦 = (𝑥𝐵)))
35 imaeq2 4884 . . . . . . . . 9 (𝑦 = (𝑥𝐵) → (𝐹𝑦) = (𝐹 “ (𝑥𝐵)))
3635eleq1d 2209 . . . . . . . 8 (𝑦 = (𝑥𝐵) → ((𝐹𝑦) ∈ 𝐽 ↔ (𝐹 “ (𝑥𝐵)) ∈ 𝐽))
3736adantl 275 . . . . . . 7 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑦 = (𝑥𝐵)) → ((𝐹𝑦) ∈ 𝐽 ↔ (𝐹 “ (𝑥𝐵)) ∈ 𝐽))
3827, 34, 37ralxfr2d 4392 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽 ↔ ∀𝑥𝐾 (𝐹 “ (𝑥𝐵)) ∈ 𝐽))
39 simplrr 526 . . . . . . . . . 10 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → 𝐹: 𝐽𝐵)
40 ffun 5282 . . . . . . . . . 10 (𝐹: 𝐽𝐵 → Fun 𝐹)
41 inpreima 5553 . . . . . . . . . 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 4909 . . . . . . . . . . . 12 (𝐹𝑥) ⊆ dom 𝐹
44 cnvimarndm 4910 . . . . . . . . . . . 12 (𝐹 “ ran 𝐹) = dom 𝐹
4543, 44sseqtrri 3136 . . . . . . . . . . 11 (𝐹𝑥) ⊆ (𝐹 “ ran 𝐹)
46 simpll2 1022 . . . . . . . . . . . 12 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → ran 𝐹𝐵)
47 imass2 4922 . . . . . . . . . . . 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, 48sstrid 3112 . . . . . . . . . 10 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ (𝐹𝐵))
50 df-ss 3088 . . . . . . . . . 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 2173 . . . . . . . 8 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → (𝐹 “ (𝑥𝐵)) = (𝐹𝑥))
5352eleq1d 2209 . . . . . . 7 ((((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) ∧ 𝑥𝐾) → ((𝐹 “ (𝑥𝐵)) ∈ 𝐽 ↔ (𝐹𝑥) ∈ 𝐽))
5453ralbidva 2434 . . . . . 6 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (∀𝑥𝐾 (𝐹 “ (𝑥𝐵)) ∈ 𝐽 ↔ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽))
55 simprr 522 . . . . . . . 8 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐹: 𝐽𝐵)
5655, 31fssd 5292 . . . . . . 7 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → 𝐹: 𝐽𝑌)
5756biantrurd 303 . . . . . 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 303 . . . . 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 521 . . . . . 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 12403 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹: 𝐽𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
6462, 28, 63syl2anc 409 . . . 4 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹: 𝐽𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
6518adantr 274 . . . . 5 (((𝐾 ∈ (TopOn‘𝑌) ∧ ran 𝐹𝐵𝐵𝑌) ∧ (𝐽 ∈ Top ∧ 𝐹: 𝐽𝐵)) → (𝐾t 𝐵) ∈ (TopOn‘𝐵))
66 iscn 12403 . . . . 5 ((𝐽 ∈ (TopOn‘ 𝐽) ∧ (𝐾t 𝐵) ∈ (TopOn‘𝐵)) → (𝐹 ∈ (𝐽 Cn (𝐾t 𝐵)) ↔ (𝐹: 𝐽𝐵 ∧ ∀𝑦 ∈ (𝐾t 𝐵)(𝐹𝑦) ∈ 𝐽)))
6762, 65, 66syl2anc 409 . . . 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 695 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 963   = wceq 1332  wcel 1481  wral 2417  wrex 2418  Vcvv 2689  cin 3074  wss 3075   cuni 3743  ccnv 4545  dom cdm 4546  ran crn 4547  cima 4549  Fun wfun 5124   Fn wfn 5125  wf 5126  cfv 5130  (class class class)co 5781  t crest 12157  Topctop 12201  TopOnctopon 12214   Cn ccn 12391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-rest 12159  df-topgen 12178  df-top 12202  df-topon 12215  df-bases 12247  df-cn 12394
This theorem is referenced by:  cnrest2r  12443  hmeores  12521
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