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Theorem iscncl 21295
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 21275 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
213expa 1112 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐹:𝑋𝑌)
3 cnclima 21294 . . . . 5 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ (Clsd‘𝐾)) → (𝐹𝑦) ∈ (Clsd‘𝐽))
43ralrimiva 3104 . . . 4 (𝐹 ∈ (𝐽 Cn 𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
54adantl 473 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
62, 5jca 555 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹 ∈ (𝐽 Cn 𝐾)) → (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽)))
7 simprl 811 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → 𝐹:𝑋𝑌)
8 toponuni 20941 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
98ad3antrrr 768 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑋 = 𝐽)
10 simplrl 819 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐹:𝑋𝑌)
11 fimacnv 6511 . . . . . . . . . . 11 (𝐹:𝑋𝑌 → (𝐹𝑌) = 𝑋)
1211eqcomd 2766 . . . . . . . . . 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 2796 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐽 = (𝐹𝑌))
1514difeq1d 3870 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) = ((𝐹𝑌) ∖ (𝐹𝑥)))
16 ffun 6209 . . . . . . . 8 (𝐹:𝑋𝑌 → Fun 𝐹)
17 funcnvcnv 6117 . . . . . . . 8 (Fun 𝐹 → Fun 𝐹)
18 imadif 6134 . . . . . . . 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 2797 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) = (𝐹 “ (𝑌𝑥)))
21 imaeq2 5620 . . . . . . . 8 (𝑦 = (𝑌𝑥) → (𝐹𝑦) = (𝐹 “ (𝑌𝑥)))
2221eleq1d 2824 . . . . . . 7 (𝑦 = (𝑌𝑥) → ((𝐹𝑦) ∈ (Clsd‘𝐽) ↔ (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽)))
23 simplrr 820 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))
24 toponuni 20941 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝑌 = 𝐾)
2524ad3antlr 769 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝑌 = 𝐾)
2625difeq1d 3870 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝑌𝑥) = ( 𝐾𝑥))
27 topontop 20940 . . . . . . . . . 10 (𝐾 ∈ (TopOn‘𝑌) → 𝐾 ∈ Top)
2827ad3antlr 769 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐾 ∈ Top)
29 eqid 2760 . . . . . . . . . 10 𝐾 = 𝐾
3029opncld 21059 . . . . . . . . 9 ((𝐾 ∈ Top ∧ 𝑥𝐾) → ( 𝐾𝑥) ∈ (Clsd‘𝐾))
3128, 30sylancom 704 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐾𝑥) ∈ (Clsd‘𝐾))
3226, 31eqeltrd 2839 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝑌𝑥) ∈ (Clsd‘𝐾))
3322, 23, 32rspcdva 3455 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹 “ (𝑌𝑥)) ∈ (Clsd‘𝐽))
3420, 33eqeltrd 2839 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽))
35 topontop 20940 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
3635ad3antrrr 768 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → 𝐽 ∈ Top)
37 cnvimass 5643 . . . . . . . 8 (𝐹𝑥) ⊆ dom 𝐹
38 fdm 6212 . . . . . . . . 9 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
3910, 38syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → dom 𝐹 = 𝑋)
4037, 39syl5sseq 3794 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝑋)
4140, 9sseqtrd 3782 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ⊆ 𝐽)
42 eqid 2760 . . . . . . 7 𝐽 = 𝐽
4342isopn2 21058 . . . . . 6 ((𝐽 ∈ Top ∧ (𝐹𝑥) ⊆ 𝐽) → ((𝐹𝑥) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽)))
4436, 41, 43syl2anc 696 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → ((𝐹𝑥) ∈ 𝐽 ↔ ( 𝐽 ∖ (𝐹𝑥)) ∈ (Clsd‘𝐽)))
4534, 44mpbird 247 . . . 4 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) ∧ 𝑥𝐾) → (𝐹𝑥) ∈ 𝐽)
4645ralrimiva 3104 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)
47 iscn 21261 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
4847adantr 472 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑥𝐾 (𝐹𝑥) ∈ 𝐽)))
497, 46, 48mpbir2and 995 . 2 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑦 ∈ (Clsd‘𝐾)(𝐹𝑦) ∈ (Clsd‘𝐽))) → 𝐹 ∈ (𝐽 Cn 𝐾))
506, 49impbida 913 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 1632  wcel 2139  wral 3050  cdif 3712  wss 3715   cuni 4588  ccnv 5265  dom cdm 5266  cima 5269  Fun wfun 6043  wf 6045  cfv 6049  (class class class)co 6814  Topctop 20920  TopOnctopon 20937  Clsdccld 21042   Cn ccn 21250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-map 8027  df-top 20921  df-topon 20938  df-cld 21045  df-cn 21253
This theorem is referenced by:  cncls2  21299  paste  21320  cmphaushmeo  21825  ubthlem1  28056  ubthlem2  28057
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