Theorem cncnpi 12868

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 2165	. . . 4 ⊢ ∪ 𝐾 = ∪ 𝐾
3	1, 2	cnf 12844	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:𝑋⟶∪ 𝐾)
4	3	adantr 274	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹:𝑋⟶∪ 𝐾)
5		cnima 12860	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
6	5	ad2ant2r 501	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (◡𝐹 “ 𝑦) ∈ 𝐽)
7		simpr 109	. . . . . . 7 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
8	7	adantr 274	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ 𝑋)
9		simprr 522	. . . . . 6 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → (𝐹‘𝐴) ∈ 𝑦)
10	3	ad2antrr 480	. . . . . . 7 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐹:𝑋⟶∪ 𝐾)
11		ffn 5337	. . . . . . 7 ⊢ (𝐹:𝑋⟶∪ 𝐾 → 𝐹 Fn 𝑋)
12		elpreima 5604	. . . . . . 7 ⊢ (𝐹 Fn 𝑋 → (𝐴 ∈ (◡𝐹 “ 𝑦) ↔ (𝐴 ∈ 𝑋 ∧ (𝐹‘𝐴) ∈ 𝑦)))
13	10, 11, 12	3syl 17	. . . . . 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 934	. . . . 5 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → 𝐴 ∈ (◡𝐹 “ 𝑦))
15		eqimss 3196	. . . . . . . 8 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → 𝑥 ⊆ (◡𝐹 “ 𝑦))
16	15	biantrud 302	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
17		eleq2 2230	. . . . . . 7 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ (◡𝐹 “ 𝑦)))
18	16, 17	bitr3d 189	. . . . . 6 ⊢ (𝑥 = (◡𝐹 “ 𝑦) → ((𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)) ↔ 𝐴 ∈ (◡𝐹 “ 𝑦)))
19	18	rspcev 2830	. . . . 5 ⊢ (((◡𝐹 “ 𝑦) ∈ 𝐽 ∧ 𝐴 ∈ (◡𝐹 “ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
20	6, 14, 19	syl2anc 409	. . . 4 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ (𝐹‘𝐴) ∈ 𝑦)) → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦)))
21	20	expr 373	. . 3 ⊢ (((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
22	21	ralrimiva 2539	. 2 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → ∀𝑦 ∈ 𝐾 ((𝐹‘𝐴) ∈ 𝑦 → ∃𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ∧ 𝑥 ⊆ (◡𝐹 “ 𝑦))))
23		cntop1 12841	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
24	23	adantr 274	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ Top)
25	1	toptopon 12656	. . . 4 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
26	24, 25	sylib 121	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
27		cntop2 12842	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
28	27	adantr 274	. . . 4 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ Top)
29	2	toptopon 12656	. . . 4 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾))
30	28, 29	sylib 121	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐾 ∈ (TopOn‘∪ 𝐾))
31		iscnp3 12843	. . 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 1228	. 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 934	1 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝐴))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445   ⊆ wss 3116  ∪ cuni 3789  ^◡ccnv 4603   “ cima 4607   Fn wfn 5183  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  Topctop 12635  TopOnctopon 12648   Cn ccn 12825   CnP ccnp 12826

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-top 12636  df-topon 12649  df-cn 12828  df-cnp 12829

This theorem is referenced by:  cnsscnp  12869  cncnp  12870  lmcn  12891  dvcnp2cntop  13303  dvaddxxbr  13305  dvmulxxbr  13306  dvcoapbr  13311  dvcjbr  13312