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Theorem iscncl 22702
Description: A characterization of a continuity 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 22682 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expa 1118 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
3 cnclima 22701 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) ∈ (Clsd‘𝐽))
43ralrimiva 3145 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
54adantl 482 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
62, 5jca 512 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽)))
7 simprl 769 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → 𝐹:𝑋𝑌)
8 toponuni 22345 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
98ad3antrrr 728 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑋 = 𝐽)
10 simplrl 775 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐹:𝑋𝑌)
11 fimacnv 6726 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
1211eqcomd 2737 . . . . . . . . . 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 2773 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐽 = (𝐹𝑌))
1514difeq1d 4117 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) = ((𝐹𝑌) ∖ (𝐹𝑥)))
16 ffun 6707 . . . . . . . 8 (𝐹:𝑋𝑌 → Fun 𝐹)
17 funcnvcnv 6604 . . . . . . . 8 (Fun 𝐹 → Fun 𝐹)
18 imadif 6621 . . . . . . . 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 2774 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) = (𝐹 “ (𝑌𝑥)))
21 imaeq2 6045 . . . . . . . 8 (𝑦 = (𝑌𝑥) → (𝐹𝑦) = (𝐹 “ (𝑌𝑥)))
2221eleq1d 2817 . . . . . . 7 (𝑦 = (𝑌𝑥) → ((𝐹𝑦) ∈ (Clsd‘𝐽) ↔ (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽)))
23 simplrr 776 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
24 toponuni 22345 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2524ad3antlr 729 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑌 = 𝐾)
2625difeq1d 4117 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝑌𝑥) = ( 𝐾𝑥))
27 topontop 22344 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2827ad3antlr 729 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐾 ∈ Top)
29 eqid 2731 . . . . . . . . . 10 𝐾 = 𝐾
3029opncld 22466 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑥𝐾) → ( 𝐾𝑥) ∈ (Clsd‘𝐾))
3128, 30sylancom 588 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐾𝑥) ∈ (Clsd‘𝐾))
3226, 31eqeltrd 2832 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝑌𝑥) ∈ (Clsd‘𝐾))
3322, 23, 32rspcdva 3610 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽))
3420, 33eqeltrd 2832 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
35 topontop 22344 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3635ad3antrrr 728 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐽 ∈ Top)
37 cnvimass 6069 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
3837, 10fssdm 6724 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝑋)
3938, 9sseqtrd 4018 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝐽)
40 eqid 2731 . . . . . . 7 𝐽 = 𝐽
4140isopn2 22465 . . . . . 6 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((𝐹𝑥) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽)))
4236, 39, 41syl2anc 584 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ((𝐹𝑥) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽)))
4334, 42mpbird 256 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ∈ 𝐽)
4443ralrimiva 3145 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)
45 iscn 22668 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
4645adantr 481 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
477, 44, 46mpbir2and 711 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → 𝐹 ∈ (𝐽 Cn 𝐾))
486, 47impbida 799 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 205  wa 396   = wceq 1541  wcel 2106  wral 3060  cdif 3941  wss 3944   cuni 4901  ccnv 5668  cima 5672  Fun wfun 6526  wf 6528  cfv 6532  (class class class)co 7393  Topctop 22324  TopOnctopon 22341  Clsdccld 22449   Cn ccn 22657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-map 8805  df-top 22325  df-topon 22342  df-cld 22452  df-cn 22660
This theorem is referenced by:  cncls2  22706  paste  22727  cmphaushmeo  23233  ubthlem1  29986  ubthlem2  29987  rhmpreimacn  32694
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