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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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