Theorem cncnpi 23251

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 2734	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 23219	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
4	3	adantr 480	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
5		cnima 23238	. . . . . 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 484	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8	7	adantr 480	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ 𝑋)
9		simprr 772	. . . . . 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 6717	. . . . . . 7 ⊢ (𝐹:𝑋⟶∪ 𝐾 → 𝐹 Fn 𝑋)
12		elpreima 7059	. . . . . . 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 4024	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → 𝑥 ⊆ (◡𝐹 “ 𝑦))
16	15	biantrud 531	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
17		eleq2 2822	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (◡𝐹 “ 𝑦)))
18	16, 17	bitr3d 281	. . . . . 6 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)) ↔ 𝐴 ∈ (◡𝐹 “ 𝑦)))
19	18	rspcev 3606	. . . . 5 ⊢ (((◡𝐹 “ 𝑦) ∈ 𝐽 ∧ 𝐴 ∈ (◡𝐹 “ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
20	6, 14, 19	syl2anc 584	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
21	20	expr 456	. . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
22	21	ralrimiva 3133	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
23		cntop1 23213	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
24	23	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
25	1	toptopon 22890	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
26	24, 25	sylib 218	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27		cntop2 23214	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
28	27	adantr 480	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ Top)
29	2	toptopon 22890	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
30	28, 29	sylib 218	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
31		iscnp3 23217	. . 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 1372	. 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 206   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059   ⊆ wss 3933  ∪ cuni 4889  ^◡ccnv 5666   “ cima 5670   Fn wfn 6537  ⟶wf 6538  ‘cfv 6542  (class class class)co 7414  Topctop 22866  TopOnctopon 22883   Cn ccn 23197   CnP ccnp 23198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5278  ax-nul 5288  ax-pow 5347  ax-pr 5414  ax-un 7738

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3421  df-v 3466  df-sbc 3773  df-dif 3936  df-un 3938  df-in 3940  df-ss 3950  df-nul 4316  df-if 4508  df-pw 4584  df-sn 4609  df-pr 4611  df-op 4615  df-uni 4890  df-br 5126  df-opab 5188  df-mpt 5208  df-id 5560  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6495  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-map 8851  df-top 22867  df-topon 22884  df-cn 23200  df-cnp 23201

This theorem is referenced by:  cnsscnp  23252  cncnp  23253  lmcn  23278  ptcn  23600  tmdcn2  24062  ghmcnp  24088  tsmsmhm  24119  tsmsadd  24120  dvcnp2  25910  dvcnp2OLD  25911  dvaddbr  25929  dvmulbr  25930  dvmulbrOLD  25931  dvcobr  25938  dvcobrOLD  25939  dvcjbr  25942  dvcnvlem  25969  lhop1lem  26007  dvcnvrelem2  26012  ftc1cn  26039  taylthlem2  26371  taylthlem2OLD  26372  psercn  26425  abelth  26440  cxpcn3  26746  efrlim  26967  efrlimOLD  26968  blocni  30771  cvmlift2lem11  35259  cvmlift2lem12  35260  cvmlift3lem7  35271  poimir  37601  ftc1cnnc  37640  cncfiooicclem1  45853  fouriercn  46192