Theorem cncnpi 23608

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 2763	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 23576	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
4	3	adantr 486	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
5		cnima 23595	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
6	5	ad2ant2r 760	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
7		simpr 490	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8	7	adantr 486	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ 𝑋)
9		simprr 785	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (𝐹‘𝐴) ∈ 𝑦)
10	3	ad2antrr 739	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐹:𝑋⟶∪ 𝐾)
11		ffn 6717	. . . . . . 7 ⊢ (𝐹:𝑋⟶∪ 𝐾 → 𝐹 Fn 𝑋)
12		elpreima 7065	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → (𝐴 ∈ (◡𝐹 “ 𝑦) ↔ (𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ 𝑦)))
13	10, 11, 12	3syl 19	. . . . . 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 726	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ (◡𝐹 “ 𝑦))
15		eqimss 3970	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → 𝑥 ⊆ (◡𝐹 “ 𝑦))
16	15	biantrud 541	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
17		eleq2 2852	. . . . . . 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 3579	. . . . 5 ⊢ (((◡𝐹 “ 𝑦) ∈ 𝐽 ∧ 𝐴 ∈ (◡𝐹 “ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
20	6, 14, 19	syl2anc 596	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
21	20	expr 462	. . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
22	21	ralrimiva 3157	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
23		cntop1 23570	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
24	23	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
25	1	toptopon 23247	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
26	24, 25	sylib 221	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27		cntop2 23571	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
28	27	adantr 486	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ Top)
29	2	toptopon 23247	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
30	28, 29	sylib 221	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
31		iscnp3 23574	. . 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 1398	. 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 726	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2146  ∀wral 3079  ∃wrex 3089   ⊆ wss 3899  ∪ cuni 4869  ^◡ccnv 5653   “ cima 5657   Fn wfn 6541  ⟶wf 6542  ‘cfv 6546  (class class class)co 7427  Topctop 23223  TopOnctopon 23240   Cn ccn 23554   CnP ccnp 23555

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-sbc 3742  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-mpt 5189  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-fv 6554  df-ov 7430  df-oprab 7431  df-mpo 7432  df-map 8860  df-top 23224  df-topon 23241  df-cn 23557  df-cnp 23558

This theorem is used by:  cnsscnp  23609  cncnp  23610  lmcn  23635  ptcn  23958  tmdcn2  24420  ghmcnp  24446  tsmsmhm  24477  tsmsadd  24478  dvcnp2  26252  dvaddbr  26270  dvmulbr  26271  dvcobr  26278  dvcjbr  26281  dvcnvlem  26308  lhop1lem  26345  dvcnvrelem2  26350  ftc1cn  26375  taylthlem2  26713  psercn  26765  abelth  26780  cxpcn3  27088  efrlim  27309  blocni  31419  cvmlift2lem11  36199  cvmlift2lem12  36200  cvmlift3lem7  36211  poimir  38711  ftc1cnnc  38750  cncfiooicclem1  47042  fouriercn  47381