Theorem cnconst2 21893

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 6570	. . 3 ⊢ (𝐵 ∈ 𝑌 → (𝑋 × {𝐵}):𝑋⟶𝑌)
2	1	3ad2ant3 1131	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}):𝑋⟶𝑌)
3	2	adantr 483	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐵}):𝑋⟶𝑌)
4		simpll3 1210	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → 𝐵 ∈ 𝑌)
5		simplr 767	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → 𝑥 ∈ 𝑋)
6		fvconst2g 6966	. . . . . . . 8 ⊢ ((𝐵 ∈ 𝑌 ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
7	4, 5, 6	syl2anc 586	. . . . . . 7 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
8	7	eleq1d 2899	. . . . . 6 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 ↔ 𝐵 ∈ 𝑦))
9		simpll1 1208	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
10		toponmax 21536	. . . . . . . . 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 767	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝑥 ∈ 𝑋)
13		df-ima 5570	. . . . . . . . 9 ⊢ ((𝑋 × {𝐵}) “ 𝑋) = ran ((𝑋 × {𝐵}) ↾ 𝑋)
14		ssid 3991	. . . . . . . . . . . . 13 ⊢ 𝑋 ⊆ 𝑋
15		xpssres 5891	. . . . . . . . . . . . 13 ⊢ (𝑋 ⊆ 𝑋 → ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵}))
16	14, 15	ax-mp 5	. . . . . . . . . . . 12 ⊢ ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵})
17	16	rneqi 5809	. . . . . . . . . . 11 ⊢ ran ((𝑋 × {𝐵}) ↾ 𝑋) = ran (𝑋 × {𝐵})
18		rnxpss 6031	. . . . . . . . . . 11 ⊢ ran (𝑋 × {𝐵}) ⊆ {𝐵}
19	17, 18	eqsstri 4003	. . . . . . . . . 10 ⊢ ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ {𝐵}
20		simprr 771	. . . . . . . . . . 11 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → 𝐵 ∈ 𝑦)
21	20	snssd 4744	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → {𝐵} ⊆ 𝑦)
22	19, 21	sstrid 3980	. . . . . . . . 9 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ 𝑦)
23	13, 22	eqsstrid 4017	. . . . . . . 8 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐵 ∈ 𝑦)) → ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)
24		eleq2 2903	. . . . . . . . . 10 ⊢ (𝑢 = 𝑋 → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ 𝑋))
25		imaeq2 5927	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑋 → ((𝑋 × {𝐵}) “ 𝑢) = ((𝑋 × {𝐵}) “ 𝑋))
26	25	sseq1d 4000	. . . . . . . . . 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 3625	. . . . . . . 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 459	. . . . . 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 242	. . . . 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 3184	. . . 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 1187	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34		simpl2 1188	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐾 ∈ (TopOn‘𝑌))
35		simpr 487	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
36		iscnp 21847	. . . . 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 1367	. . . 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 711	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) ∧ 𝑥 ∈ 𝑋) → (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
39	38	ralrimiva 3184	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → ∀𝑥 ∈ 𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
40		cncnp 21890	. . 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 1128	. 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 711	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵 ∈ 𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ⊆ wss 3938  {csn 4569   × cxp 5555  ran crn 5558   ↾ cres 5559   “ cima 5560  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  TopOnctopon 21520   Cn ccn 21834   CnP ccnp 21835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-1st 7691  df-2nd 7692  df-map 8410  df-topgen 16719  df-top 21504  df-topon 21521  df-cn 21837  df-cnp 21838

This theorem is referenced by:  cnconst  21894  xkoccn  22229  txkgen  22262  cnmptc  22272  pcoptcl  23627  blocni  28584  pl1cn  31200  connpconn  32484  cvmliftphtlem  32566  cvmlift3lem9  32576  cnfdmsn  42172  stoweidlem47  42339