Theorem cnconst2 21087

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 6094	. . 3 ⊢ (𝐵 ∈ 𝑌 → (𝑋 × {𝐵}):𝑋⟶𝑌)
2	1	3ad2ant3 1084	. 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 1102	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → 𝐵 ∈ 𝑌)
5		simplr 792	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → 𝑥 ∈ 𝑋)
6		fvconst2g 6467	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
7	4, 5, 6	syl2anc 693	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
8	7	eleq1d 2686	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 ↔ 𝐵 ∈ 𝑦))
9		simpll1 1100	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
10		toponmax 20730	. . . . . . . . 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 792	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝑥 ∈ 𝑋)
13		df-ima 5127	. . . . . . . . 9 ⊢ ((𝑋 × {𝐵}) “ 𝑋) = ran ((𝑋 × {𝐵}) ↾ 𝑋)
14		ssid 3624	. . . . . . . . . . . . 13 ⊢ 𝑋 ⊆ 𝑋
15		xpssres 5434	. . . . . . . . . . . . 13 ⊢ (𝑋 ⊆ 𝑋 → ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵}))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵})
17	16	rneqi 5352	. . . . . . . . . . 11 ⊢ ran ((𝑋 × {𝐵}) ↾ 𝑋) = ran (𝑋 × {𝐵})
18		rnxpss 5566	. . . . . . . . . . 11 ⊢ ran (𝑋 × {𝐵}) ⊆ {𝐵}
19	17, 18	eqsstri 3635	. . . . . . . . . 10 ⊢ ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ {𝐵}
20		simprr 796	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝐵 ∈ 𝑦)
21	20	snssd 4340	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → {𝐵} ⊆ 𝑦)
22	19, 21	syl5ss 3614	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ 𝑦)
23	13, 22	syl5eqss 3649	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)
24		eleq2 2690	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑋))
25		imaeq2 5462	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑋 → ((𝑋 × {𝐵}) “ 𝑢) = ((𝑋 × {𝐵}) “ 𝑋))
26	25	sseq1d 3632	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦 ↔ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦))
27	24, 26	anbi12d 747	. . . . . . . . 9 ⊢ (𝑢 = 𝑋 → ((𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)))
28	27	rspcev 3309	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐽 ∧ (𝑥 ∈ 𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)) → ∃𝑢 ∈ 𝐽 (𝑥 ∈ 𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
29	11, 12, 23, 28	syl12anc 1324	. . . . . . 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 643	. . . . . 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 230	. . . . 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 2966	. . . 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 1064	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34		simpl2 1065	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
35		simpr 477	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
36		iscnp 21041	. . . . 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 1326	. . . 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 957	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
39	38	ralrimiva 2966	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
40		cncnp 21084	. . 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 1081	. 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 957	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  {csn 4177   × cxp 5112  ran crn 5115   ↾ cres 5116   “ cima 5117  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  TopOnctopon 20715   Cn ccn 21028   CnP ccnp 21029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-topgen 16104  df-top 20699  df-topon 20716  df-cn 21031  df-cnp 21032

This theorem is referenced by:  cnconst  21088  xkoccn  21422  txkgen  21455  cnmptc  21465  pcoptcl  22821  blocni  27660  pl1cn  30001  connpconn  31217  cvmliftphtlem  31299  cvmlift3lem9  31309  cnfdmsn  40095  stoweidlem47  40264