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Theorem iscncl 21805
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 21785 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expa 1110 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
3 cnclima 21804 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) ∈ (Clsd‘𝐽))
43ralrimiva 3179 . . . 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 767 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → 𝐹:𝑋𝑌)
8 toponuni 21450 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
98ad3antrrr 726 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑋 = 𝐽)
10 simplrl 773 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐹:𝑋𝑌)
11 fimacnv 6831 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
1211eqcomd 2824 . . . . . . . . . 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 2855 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐽 = (𝐹𝑌))
1514difeq1d 4095 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) = ((𝐹𝑌) ∖ (𝐹𝑥)))
16 ffun 6510 . . . . . . . 8 (𝐹:𝑋𝑌 → Fun 𝐹)
17 funcnvcnv 6414 . . . . . . . 8 (Fun 𝐹 → Fun 𝐹)
18 imadif 6431 . . . . . . . 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 2856 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) = (𝐹 “ (𝑌𝑥)))
21 imaeq2 5918 . . . . . . . 8 (𝑦 = (𝑌𝑥) → (𝐹𝑦) = (𝐹 “ (𝑌𝑥)))
2221eleq1d 2894 . . . . . . 7 (𝑦 = (𝑌𝑥) → ((𝐹𝑦) ∈ (Clsd‘𝐽) ↔ (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽)))
23 simplrr 774 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
24 toponuni 21450 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2524ad3antlr 727 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑌 = 𝐾)
2625difeq1d 4095 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝑌𝑥) = ( 𝐾𝑥))
27 topontop 21449 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2827ad3antlr 727 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐾 ∈ Top)
29 eqid 2818 . . . . . . . . . 10 𝐾 = 𝐾
3029opncld 21569 . . . . . . . . 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 2910 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝑌𝑥) ∈ (Clsd‘𝐾))
3322, 23, 32rspcdva 3622 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽))
3420, 33eqeltrd 2910 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
35 topontop 21449 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3635ad3antrrr 726 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐽 ∈ Top)
37 cnvimass 5942 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
3837, 10fssdm 6523 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝑋)
3938, 9sseqtrd 4004 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝐽)
40 eqid 2818 . . . . . . 7 𝐽 = 𝐽
4140isopn2 21568 . . . . . 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 258 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ∈ 𝐽)
4443ralrimiva 3179 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)
45 iscn 21771 . . . 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 709 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → 𝐹 ∈ (𝐽 Cn 𝐾))
486, 47impbida 797 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  cdif 3930  wss 3933   cuni 4830  ccnv 5547  cima 5551  Fun wfun 6342  wf 6344  cfv 6348  (class class class)co 7145  Topctop 21429  TopOnctopon 21446  Clsdccld 21552   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-top 21430  df-topon 21447  df-cld 21555  df-cn 21763
This theorem is referenced by:  cncls2  21809  paste  21830  cmphaushmeo  22336  ubthlem1  28574  ubthlem2  28575
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