Theorem cnconst2 22434

Description: A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
cnconst2	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))

Proof of Theorem cnconst2

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

Step	Hyp	Ref	Expression
1		fconst6g 6663	. . 3 ⊢ (𝐵 ∈ 𝑌 → (𝑋 × {𝐵}):𝑋⟶𝑌)
2	1	3ad2ant3 1134	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}):𝑋⟶𝑌)
3	2	adantr 481	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐵}):𝑋⟶𝑌)
4		simpll3 1213	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → 𝐵 ∈ 𝑌)
5		simplr 766	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → 𝑥 ∈ 𝑋)
6		fvconst2g 7077	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
7	4, 5, 6	syl2anc 584	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
8	7	eleq1d 2823	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 ↔ 𝐵 ∈ 𝑦))
9		simpll1 1211	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
10		toponmax 22075	. . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽)
11	9, 10	syl 17	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝑋 ∈ 𝐽)
12		simplr 766	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝑥 ∈ 𝑋)
13		df-ima 5602	. . . . . . . . 9 ⊢ ((𝑋 × {𝐵}) “ 𝑋) = ran ((𝑋 × {𝐵}) ↾ 𝑋)
14		ssid 3943	. . . . . . . . . . . . 13 ⊢ 𝑋 ⊆ 𝑋
15		xpssres 5928	. . . . . . . . . . . . 13 ⊢ (𝑋 ⊆ 𝑋 → ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵}))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵})
17	16	rneqi 5846	. . . . . . . . . . 11 ⊢ ran ((𝑋 × {𝐵}) ↾ 𝑋) = ran (𝑋 × {𝐵})
18		rnxpss 6075	. . . . . . . . . . 11 ⊢ ran (𝑋 × {𝐵}) ⊆ {𝐵}
19	17, 18	eqsstri 3955	. . . . . . . . . 10 ⊢ ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ {𝐵}
20		simprr 770	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝐵 ∈ 𝑦)
21	20	snssd 4742	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → {𝐵} ⊆ 𝑦)
22	19, 21	sstrid 3932	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ 𝑦)
23	13, 22	eqsstrid 3969	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)
24		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑋))
25		imaeq2 5965	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑋 → ((𝑋 × {𝐵}) “ 𝑢) = ((𝑋 × {𝐵}) “ 𝑋))
26	25	sseq1d 3952	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦 ↔ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦))
27	24, 26	anbi12d 631	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → ((𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)))
28	27	rspcev 3561	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑥 ∈ 𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
29	11, 12, 23, 28	syl12anc 834	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
30	29	expr 457	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → (𝐵 ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
31	8, 30	sylbid 239	. . . . 5 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
32	31	ralrimiva 3103	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → ∀𝑦 ∈ 𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
33		simpl1 1190	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34		simpl2 1191	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
35		simpr 485	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
36		iscnp 22388	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ((𝑋 × {𝐵}):𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))))
37	33, 34, 35, 36	syl3anc 1370	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ((𝑋 × {𝐵}):𝑋⟶𝑌 ∧ ∀𝑦 ∈ 𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))))
38	3, 32, 37	mpbir2and 710	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
39	38	ralrimiva 3103	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
40		cncnp 22431	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 × {𝐵}):𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))))
41	40	3adant3 1131	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → ((𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 × {𝐵}):𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))))
42	2, 39, 41	mpbir2and 710	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  {csn 4561   × cxp 5587  ran crn 5590   ↾ cres 5591   “ cima 5592  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  TopOnctopon 22059   Cn ccn 22375   CnP ccnp 22376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-topgen 17154  df-top 22043  df-topon 22060  df-cn 22378  df-cnp 22379

This theorem is referenced by:  cnconst  22435  xkoccn  22770  txkgen  22803  cnmptc  22813  pcoptcl  24184  blocni  29167  pl1cn  31905  connpconn  33197  cvmliftphtlem  33279  cvmlift3lem9  33289  cnfdmsn  43423  stoweidlem47  43588