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Theorem cnconst2 14401
Description: A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
cnconst2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))

Proof of Theorem cnconst2
Dummy variables 𝑥 𝑢 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fconst6g 5452 . . 3 (𝐵𝑌 → (𝑋 × {𝐵}):𝑋𝑌)
213ad2ant3 1022 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}):𝑋𝑌)
32adantr 276 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → (𝑋 × {𝐵}):𝑋𝑌)
4 simpll3 1040 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → 𝐵𝑌)
5 simplr 528 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → 𝑥𝑋)
6 fvconst2g 5772 . . . . . . . 8 ((𝐵𝑌𝑥𝑋) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
74, 5, 6syl2anc 411 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
87eleq1d 2262 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦𝐵𝑦))
9 simpll1 1038 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
10 toponmax 14193 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
119, 10syl 14 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝑋𝐽)
12 simplr 528 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝑥𝑋)
13 df-ima 4672 . . . . . . . . 9 ((𝑋 × {𝐵}) “ 𝑋) = ran ((𝑋 × {𝐵}) ↾ 𝑋)
14 ssid 3199 . . . . . . . . . . . . 13 𝑋𝑋
15 xpssres 4977 . . . . . . . . . . . . 13 (𝑋𝑋 → ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵}))
1614, 15ax-mp 5 . . . . . . . . . . . 12 ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵})
1716rneqi 4890 . . . . . . . . . . 11 ran ((𝑋 × {𝐵}) ↾ 𝑋) = ran (𝑋 × {𝐵})
18 rnxpss 5097 . . . . . . . . . . 11 ran (𝑋 × {𝐵}) ⊆ {𝐵}
1917, 18eqsstri 3211 . . . . . . . . . 10 ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ {𝐵}
20 simprr 531 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝐵𝑦)
2120snssd 3763 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → {𝐵} ⊆ 𝑦)
2219, 21sstrid 3190 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ 𝑦)
2313, 22eqsstrid 3225 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)
24 eleq2 2257 . . . . . . . . . 10 (𝑢 = 𝑋 → (𝑥𝑢𝑥𝑋))
25 imaeq2 5001 . . . . . . . . . . 11 (𝑢 = 𝑋 → ((𝑋 × {𝐵}) “ 𝑢) = ((𝑋 × {𝐵}) “ 𝑋))
2625sseq1d 3208 . . . . . . . . . 10 (𝑢 = 𝑋 → (((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦 ↔ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦))
2724, 26anbi12d 473 . . . . . . . . 9 (𝑢 = 𝑋 → ((𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦) ↔ (𝑥𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)))
2827rspcev 2864 . . . . . . . 8 ((𝑋𝐽 ∧ (𝑥𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
2911, 12, 23, 28syl12anc 1247 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
3029expr 375 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (𝐵𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
318, 30sylbid 150 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
3231ralrimiva 2567 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → ∀𝑦𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
33 simpl1 1002 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34 simpl2 1003 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
35 simpr 110 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝑥𝑋)
36 iscnp 14367 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥𝑋) → ((𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ((𝑋 × {𝐵}):𝑋𝑌 ∧ ∀𝑦𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))))
3733, 34, 35, 36syl3anc 1249 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → ((𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ((𝑋 × {𝐵}):𝑋𝑌 ∧ ∀𝑦𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))))
383, 32, 37mpbir2and 946 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
3938ralrimiva 2567 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
40 cncnp 14398 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 × {𝐵}):𝑋𝑌 ∧ ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))))
41403adant3 1019 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → ((𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 × {𝐵}):𝑋𝑌 ∧ ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))))
422, 39, 41mpbir2and 946 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  wcel 2164  wral 2472  wrex 2473  wss 3153  {csn 3618   × cxp 4657  ran crn 4660  cres 4661  cima 4662  wf 5250  cfv 5254  (class class class)co 5918  TopOnctopon 14178   Cn ccn 14353   CnP ccnp 14354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-topgen 12871  df-top 14166  df-topon 14179  df-cn 14356  df-cnp 14357
This theorem is referenced by:  cnconst  14402  cnmptc  14450
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