Theorem cncnpi 22129

Description: A continuous function is continuous at all points. One direction of Theorem 7.2(g) of [Munkres] p. 107. (Contributed by Raph Levien, 20-Nov-2006.) (Proof shortened by Mario Carneiro, 21-Aug-2015.)

Hypothesis

Ref	Expression
cnsscnp.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
cncnpi	⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))

Proof of Theorem cncnpi

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnsscnp.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2		eqid 2736	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 22097	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
4	3	adantr 484	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
5		cnima 22116	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
6	5	ad2ant2r 747	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
7		simpr 488	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8	7	adantr 484	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ 𝑋)
9		simprr 773	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (𝐹‘𝐴) ∈ 𝑦)
10	3	ad2antrr 726	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐹:𝑋⟶∪ 𝐾)
11		ffn 6523	. . . . . . 7 ⊢ (𝐹:𝑋⟶∪ 𝐾 → 𝐹 Fn 𝑋)
12		elpreima 6856	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → (𝐴 ∈ (◡𝐹 “ 𝑦) ↔ (𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ 𝑦)))
13	10, 11, 12	3syl 18	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (𝐴 ∈ (◡𝐹 “ 𝑦) ↔ (𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ 𝑦)))
14	8, 9, 13	mpbir2and 713	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ (◡𝐹 “ 𝑦))
15		eqimss 3943	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → 𝑥 ⊆ (◡𝐹 “ 𝑦))
16	15	biantrud 535	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
17		eleq2 2819	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (◡𝐹 “ 𝑦)))
18	16, 17	bitr3d 284	. . . . . 6 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)) ↔ 𝐴 ∈ (◡𝐹 “ 𝑦)))
19	18	rspcev 3527	. . . . 5 ⊢ (((◡𝐹 “ 𝑦) ∈ 𝐽 ∧ 𝐴 ∈ (◡𝐹 “ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
20	6, 14, 19	syl2anc 587	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
21	20	expr 460	. . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
22	21	ralrimiva 3095	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
23		cntop1 22091	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
24	23	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
25	1	toptopon 21768	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
26	24, 25	sylib 221	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27		cntop2 22092	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
28	27	adantr 484	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ Top)
29	2	toptopon 21768	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
30	28, 29	sylib 221	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
31		iscnp3 22095	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘∪ 𝐾) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))
32	26, 30, 7, 31	syl3anc 1373	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴) ↔ (𝐹:𝑋⟶∪ 𝐾 ∧ ∀𝑦 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))))
33	4, 22, 32	mpbir2and 713	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052   ⊆ wss 3853  ∪ cuni 4805  ^◡ccnv 5535   “ cima 5539   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191  Topctop 21744  TopOnctopon 21761   Cn ccn 22075   CnP ccnp 22076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-map 8488  df-top 21745  df-topon 21762  df-cn 22078  df-cnp 22079

This theorem is referenced by:  cnsscnp  22130  cncnp  22131  lmcn  22156  ptcn  22478  tmdcn2  22940  ghmcnp  22966  tsmsmhm  22997  tsmsadd  22998  dvcnp2  24771  dvaddbr  24789  dvmulbr  24790  dvcobr  24797  dvcjbr  24800  dvcnvlem  24827  lhop1lem  24864  dvcnvrelem2  24869  ftc1cn  24894  taylthlem2  25220  psercn  25272  abelth  25287  cxpcn3  25588  efrlim  25806  blocni  28840  cvmlift2lem11  32942  cvmlift2lem12  32943  cvmlift3lem7  32954  poimir  35496  ftc1cnnc  35535  cncfiooicclem1  43052  fouriercn  43391