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Theorem iscncl 21121
 Description: A definition of a continuous function using closed sets. Theorem 1 (d) of [BourbakiTop1] p. I.9. (Contributed by FL, 19-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)
Assertion
Ref Expression
iscncl ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
Distinct variable groups:   𝑦,𝐹   𝑦,𝐽   𝑦,𝐾   𝑦,𝑋   𝑦,𝑌

Proof of Theorem iscncl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 cnf2 21101 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expa 1284 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
3 cnclima 21120 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) ∈ (Clsd‘𝐽))
43ralrimiva 2995 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
54adantl 481 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
62, 5jca 553 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽)))
7 simprl 809 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → 𝐹:𝑋𝑌)
8 toponuni 20767 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
98ad3antrrr 766 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑋 = 𝐽)
10 simplrl 817 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐹:𝑋𝑌)
11 fimacnv 6387 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
1211eqcomd 2657 . . . . . . . . . 10 (𝐹:𝑋𝑌𝑋 = (𝐹𝑌))
1310, 12syl 17 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑋 = (𝐹𝑌))
149, 13eqtr3d 2687 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐽 = (𝐹𝑌))
1514difeq1d 3760 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) = ((𝐹𝑌) ∖ (𝐹𝑥)))
16 ffun 6086 . . . . . . . 8 (𝐹:𝑋𝑌 → Fun 𝐹)
17 funcnvcnv 5994 . . . . . . . 8 (Fun 𝐹 → Fun 𝐹)
18 imadif 6011 . . . . . . . 8 (Fun 𝐹 → (𝐹 “ (𝑌𝑥)) = ((𝐹𝑌) ∖ (𝐹𝑥)))
1910, 16, 17, 184syl 19 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹 “ (𝑌𝑥)) = ((𝐹𝑌) ∖ (𝐹𝑥)))
2015, 19eqtr4d 2688 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) = (𝐹 “ (𝑌𝑥)))
21 toponuni 20767 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2221ad3antlr 767 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑌 = 𝐾)
2322difeq1d 3760 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝑌𝑥) = ( 𝐾𝑥))
24 topontop 20766 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2524ad3antlr 767 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐾 ∈ Top)
26 eqid 2651 . . . . . . . . . 10 𝐾 = 𝐾
2726opncld 20885 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑥𝐾) → ( 𝐾𝑥) ∈ (Clsd‘𝐾))
2825, 27sylancom 702 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐾𝑥) ∈ (Clsd‘𝐾))
2923, 28eqeltrd 2730 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝑌𝑥) ∈ (Clsd‘𝐾))
30 simplrr 818 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
31 imaeq2 5497 . . . . . . . . 9 (𝑦 = (𝑌𝑥) → (𝐹𝑦) = (𝐹 “ (𝑌𝑥)))
3231eleq1d 2715 . . . . . . . 8 (𝑦 = (𝑌𝑥) → ((𝐹𝑦) ∈ (Clsd‘𝐽) ↔ (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽)))
3332rspcv 3336 . . . . . . 7 ((𝑌𝑥) ∈ (Clsd‘𝐾) → (∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽) → (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽)))
3429, 30, 33sylc 65 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽))
3520, 34eqeltrd 2730 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
36 topontop 20766 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3736ad3antrrr 766 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐽 ∈ Top)
38 cnvimass 5520 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
39 fdm 6089 . . . . . . . . 9 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
4010, 39syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → dom 𝐹 = 𝑋)
4138, 40syl5sseq 3686 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝑋)
4241, 9sseqtrd 3674 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝐽)
43 eqid 2651 . . . . . . 7 𝐽 = 𝐽
4443isopn2 20884 . . . . . 6 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((𝐹𝑥) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽)))
4537, 42, 44syl2anc 694 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ((𝐹𝑥) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽)))
4635, 45mpbird 247 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ∈ 𝐽)
4746ralrimiva 2995 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)
48 iscn 21087 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
4948adantr 480 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
507, 47, 49mpbir2and 977 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → 𝐹 ∈ (𝐽 Cn 𝐾))
516, 50impbida 895 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941   ∖ cdif 3604   ⊆ wss 3607  ∪ cuni 4468  ◡ccnv 5142  dom cdm 5143   “ cima 5146  Fun wfun 5920  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  Topctop 20746  TopOnctopon 20763  Clsdccld 20868   Cn ccn 21076 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-top 20747  df-topon 20764  df-cld 20871  df-cn 21079 This theorem is referenced by:  cncls2  21125  paste  21146  cmphaushmeo  21651  ubthlem1  27854  ubthlem2  27855
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