Theorem cnconst2 23177

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 6752	. . 3 ⊢ (𝐵 ∈ 𝑌 → (𝑋 × {𝐵}):𝑋⟶𝑌)
2	1	3ad2ant3 1135	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}):𝑋⟶𝑌)
3	2	adantr 480	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐵}):𝑋⟶𝑌)
4		simpll3 1215	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → 𝐵 ∈ 𝑌)
5		simplr 768	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → 𝑥 ∈ 𝑋)
6		fvconst2g 7179	. . . . . . . 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 2814	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 ↔ 𝐵 ∈ 𝑦))
9		simpll1 1213	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
10		toponmax 22820	. . . . . . . . 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 768	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝑥 ∈ 𝑋)
13		df-ima 5654	. . . . . . . . 9 ⊢ ((𝑋 × {𝐵}) “ 𝑋) = ran ((𝑋 × {𝐵}) ↾ 𝑋)
14		ssid 3972	. . . . . . . . . . . . 13 ⊢ 𝑋 ⊆ 𝑋
15		xpssres 5992	. . . . . . . . . . . . 13 ⊢ (𝑋 ⊆ 𝑋 → ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵}))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵})
17	16	rneqi 5904	. . . . . . . . . . 11 ⊢ ran ((𝑋 × {𝐵}) ↾ 𝑋) = ran (𝑋 × {𝐵})
18		rnxpss 6148	. . . . . . . . . . 11 ⊢ ran (𝑋 × {𝐵}) ⊆ {𝐵}
19	17, 18	eqsstri 3996	. . . . . . . . . 10 ⊢ ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ {𝐵}
20		simprr 772	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝐵 ∈ 𝑦)
21	20	snssd 4776	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → {𝐵} ⊆ 𝑦)
22	19, 21	sstrid 3961	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ 𝑦)
23	13, 22	eqsstrid 3988	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)
24		eleq2 2818	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑋))
25		imaeq2 6030	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑋 → ((𝑋 × {𝐵}) “ 𝑢) = ((𝑋 × {𝐵}) “ 𝑋))
26	25	sseq1d 3981	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦 ↔ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦))
27	24, 26	anbi12d 632	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → ((𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)))
28	27	rspcev 3591	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑥 ∈ 𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
29	11, 12, 23, 28	syl12anc 836	. . . . . . 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 456	. . . . . 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 240	. . . . 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 3126	. . . 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 1192	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34		simpl2 1193	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
35		simpr 484	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
36		iscnp 23131	. . . . 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 1373	. . . 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 713	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
39	38	ralrimiva 3126	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
40		cncnp 23174	. . 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 1132	. 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 713	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  {csn 4592   × cxp 5639  ran crn 5642   ↾ cres 5643   “ cima 5644  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  TopOnctopon 22804   Cn ccn 23118   CnP ccnp 23119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804  df-topgen 17413  df-top 22788  df-topon 22805  df-cn 23121  df-cnp 23122

This theorem is referenced by:  cnconst  23178  xkoccn  23513  txkgen  23546  cnmptc  23556  pcoptcl  24928  blocni  30741  pl1cn  33952  connpconn  35229  cvmliftphtlem  35311  cvmlift3lem9  35321  cnfdmsn  45887  stoweidlem47  46052