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Theorem cnconst2 21888
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 6542 . . 3 (𝐵𝑌 → (𝑋 × {𝐵}):𝑋𝑌)
213ad2ant3 1132 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}):𝑋𝑌)
32adantr 484 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → (𝑋 × {𝐵}):𝑋𝑌)
4 simpll3 1211 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → 𝐵𝑌)
5 simplr 768 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → 𝑥𝑋)
6 fvconst2g 6941 . . . . . . . 8 ((𝐵𝑌𝑥𝑋) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
74, 5, 6syl2anc 587 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
87eleq1d 2874 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦𝐵𝑦))
9 simpll1 1209 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
10 toponmax 21531 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
119, 10syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝑋𝐽)
12 simplr 768 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝑥𝑋)
13 df-ima 5532 . . . . . . . . 9 ((𝑋 × {𝐵}) “ 𝑋) = ran ((𝑋 × {𝐵}) ↾ 𝑋)
14 ssid 3937 . . . . . . . . . . . . 13 𝑋𝑋
15 xpssres 5855 . . . . . . . . . . . . 13 (𝑋𝑋 → ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵}))
1614, 15ax-mp 5 . . . . . . . . . . . 12 ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵})
1716rneqi 5771 . . . . . . . . . . 11 ran ((𝑋 × {𝐵}) ↾ 𝑋) = ran (𝑋 × {𝐵})
18 rnxpss 5996 . . . . . . . . . . 11 ran (𝑋 × {𝐵}) ⊆ {𝐵}
1917, 18eqsstri 3949 . . . . . . . . . 10 ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ {𝐵}
20 simprr 772 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝐵𝑦)
2120snssd 4702 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → {𝐵} ⊆ 𝑦)
2219, 21sstrid 3926 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ 𝑦)
2313, 22eqsstrid 3963 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)
24 eleq2 2878 . . . . . . . . . 10 (𝑢 = 𝑋 → (𝑥𝑢𝑥𝑋))
25 imaeq2 5892 . . . . . . . . . . 11 (𝑢 = 𝑋 → ((𝑋 × {𝐵}) “ 𝑢) = ((𝑋 × {𝐵}) “ 𝑋))
2625sseq1d 3946 . . . . . . . . . 10 (𝑢 = 𝑋 → (((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦 ↔ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦))
2724, 26anbi12d 633 . . . . . . . . 9 (𝑢 = 𝑋 → ((𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦) ↔ (𝑥𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)))
2827rspcev 3571 . . . . . . . 8 ((𝑋𝐽 ∧ (𝑥𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
2911, 12, 23, 28syl12anc 835 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
3029expr 460 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (𝐵𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
318, 30sylbid 243 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
3231ralrimiva 3149 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → ∀𝑦𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
33 simpl1 1188 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34 simpl2 1189 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
35 simpr 488 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝑥𝑋)
36 iscnp 21842 . . . . 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 1368 . . . 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 712 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
3938ralrimiva 3149 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
40 cncnp 21885 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 × {𝐵}):𝑋𝑌 ∧ ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))))
41403adant3 1129 . 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 712 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2111  wral 3106  wrex 3107  wss 3881  {csn 4525   × cxp 5517  ran crn 5520  cres 5521  cima 5522  wf 6320  cfv 6324  (class class class)co 7135  TopOnctopon 21515   Cn ccn 21829   CnP ccnp 21830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-map 8391  df-topgen 16709  df-top 21499  df-topon 21516  df-cn 21832  df-cnp 21833
This theorem is referenced by:  cnconst  21889  xkoccn  22224  txkgen  22257  cnmptc  22267  pcoptcl  23626  blocni  28588  pl1cn  31308  connpconn  32595  cvmliftphtlem  32677  cvmlift3lem9  32687  cnfdmsn  42522  stoweidlem47  42687
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