Theorem cncnpi 23273

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 2726	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 23241	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
4	3	adantr 479	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
5		cnima 23260	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
6	5	ad2ant2r 745	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
7		simpr 483	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8	7	adantr 479	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ 𝑋)
9		simprr 771	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (𝐹‘𝐴) ∈ 𝑦)
10	3	ad2antrr 724	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐹:𝑋⟶∪ 𝐾)
11		ffn 6728	. . . . . . 7 ⊢ (𝐹:𝑋⟶∪ 𝐾 → 𝐹 Fn 𝑋)
12		elpreima 7071	. . . . . . 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 711	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ (◡𝐹 “ 𝑦))
15		eqimss 4038	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → 𝑥 ⊆ (◡𝐹 “ 𝑦))
16	15	biantrud 530	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
17		eleq2 2815	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (◡𝐹 “ 𝑦)))
18	16, 17	bitr3d 280	. . . . . 6 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)) ↔ 𝐴 ∈ (◡𝐹 “ 𝑦)))
19	18	rspcev 3608	. . . . 5 ⊢ (((◡𝐹 “ 𝑦) ∈ 𝐽 ∧ 𝐴 ∈ (◡𝐹 “ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
20	6, 14, 19	syl2anc 582	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
21	20	expr 455	. . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
22	21	ralrimiva 3136	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
23		cntop1 23235	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
24	23	adantr 479	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
25	1	toptopon 22910	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
26	24, 25	sylib 217	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27		cntop2 23236	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
28	27	adantr 479	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ Top)
29	2	toptopon 22910	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
30	28, 29	sylib 217	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
31		iscnp3 23239	. . 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 1368	. 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 711	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051  ∃wrex 3060   ⊆ wss 3947  ∪ cuni 4913  ^◡ccnv 5681   “ cima 5685   Fn wfn 6549  ⟶wf 6550  ‘cfv 6554  (class class class)co 7424  Topctop 22886  TopOnctopon 22903   Cn ccn 23219   CnP ccnp 23220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8857  df-top 22887  df-topon 22904  df-cn 23222  df-cnp 23223

This theorem is referenced by:  cnsscnp  23274  cncnp  23275  lmcn  23300  ptcn  23622  tmdcn2  24084  ghmcnp  24110  tsmsmhm  24141  tsmsadd  24142  dvcnp2  25940  dvcnp2OLD  25941  dvaddbr  25959  dvmulbr  25960  dvmulbrOLD  25961  dvcobr  25968  dvcobrOLD  25969  dvcjbr  25972  dvcnvlem  25999  lhop1lem  26037  dvcnvrelem2  26042  ftc1cn  26069  taylthlem2  26402  taylthlem2OLD  26403  psercn  26456  abelth  26471  cxpcn3  26776  efrlim  26997  efrlimOLD  26998  blocni  30738  cvmlift2lem11  35141  cvmlift2lem12  35142  cvmlift3lem7  35153  poimir  37354  ftc1cnnc  37393  cncfiooicclem1  45514  fouriercn  45853