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Theorem cnconst2 21135
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 6132 . . 3 (𝐵𝑌 → (𝑋 × {𝐵}):𝑋𝑌)
213ad2ant3 1104 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}):𝑋𝑌)
32adantr 480 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → (𝑋 × {𝐵}):𝑋𝑌)
4 simpll3 1122 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → 𝐵𝑌)
5 simplr 807 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → 𝑥𝑋)
6 fvconst2g 6508 . . . . . . . 8 ((𝐵𝑌𝑥𝑋) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
74, 5, 6syl2anc 694 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
87eleq1d 2715 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦𝐵𝑦))
9 simpll1 1120 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
10 toponmax 20778 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
119, 10syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝑋𝐽)
12 simplr 807 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝑥𝑋)
13 df-ima 5156 . . . . . . . . 9 ((𝑋 × {𝐵}) “ 𝑋) = ran ((𝑋 × {𝐵}) ↾ 𝑋)
14 ssid 3657 . . . . . . . . . . . . 13 𝑋𝑋
15 xpssres 5469 . . . . . . . . . . . . 13 (𝑋𝑋 → ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵}))
1614, 15ax-mp 5 . . . . . . . . . . . 12 ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵})
1716rneqi 5384 . . . . . . . . . . 11 ran ((𝑋 × {𝐵}) ↾ 𝑋) = ran (𝑋 × {𝐵})
18 rnxpss 5601 . . . . . . . . . . 11 ran (𝑋 × {𝐵}) ⊆ {𝐵}
1917, 18eqsstri 3668 . . . . . . . . . 10 ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ {𝐵}
20 simprr 811 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝐵𝑦)
2120snssd 4372 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → {𝐵} ⊆ 𝑦)
2219, 21syl5ss 3647 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ 𝑦)
2313, 22syl5eqss 3682 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)
24 eleq2 2719 . . . . . . . . . 10 (𝑢 = 𝑋 → (𝑥𝑢𝑥𝑋))
25 imaeq2 5497 . . . . . . . . . . 11 (𝑢 = 𝑋 → ((𝑋 × {𝐵}) “ 𝑢) = ((𝑋 × {𝐵}) “ 𝑋))
2625sseq1d 3665 . . . . . . . . . 10 (𝑢 = 𝑋 → (((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦 ↔ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦))
2724, 26anbi12d 747 . . . . . . . . 9 (𝑢 = 𝑋 → ((𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦) ↔ (𝑥𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)))
2827rspcev 3340 . . . . . . . 8 ((𝑋𝐽 ∧ (𝑥𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
2911, 12, 23, 28syl12anc 1364 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
3029expr 642 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (𝐵𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
318, 30sylbid 230 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
3231ralrimiva 2995 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → ∀𝑦𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
33 simpl1 1084 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34 simpl2 1085 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
35 simpr 476 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝑥𝑋)
36 iscnp 21089 . . . . 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 1366 . . . 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 977 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
3938ralrimiva 2995 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
40 cncnp 21132 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 × {𝐵}):𝑋𝑌 ∧ ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))))
41403adant3 1101 . 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 977 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607  {csn 4210   × cxp 5141  ran crn 5144  cres 5145  cima 5146  wf 5922  cfv 5926  (class class class)co 6690  TopOnctopon 20763   Cn ccn 21076   CnP ccnp 21077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-topgen 16151  df-top 20747  df-topon 20764  df-cn 21079  df-cnp 21080
This theorem is referenced by:  cnconst  21136  xkoccn  21470  txkgen  21503  cnmptc  21513  pcoptcl  22867  blocni  27788  pl1cn  30129  connpconn  31343  cvmliftphtlem  31425  cvmlift3lem9  31435  cnfdmsn  40413  stoweidlem47  40582
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