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Theorem cnconst2 22586
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 6729 . . 3 (𝐵𝑌 → (𝑋 × {𝐵}):𝑋𝑌)
213ad2ant3 1136 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → (𝑋 × {𝐵}):𝑋𝑌)
32adantr 482 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → (𝑋 × {𝐵}):𝑋𝑌)
4 simpll3 1215 . . . . . . . 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 7148 . . . . . . . 8 ((𝐵𝑌𝑥𝑋) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
74, 5, 6syl2anc 585 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → ((𝑋 × {𝐵})‘𝑥) = 𝐵)
87eleq1d 2823 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦𝐵𝑦))
9 simpll1 1213 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝐽 ∈ (TopOn‘𝑋))
10 toponmax 22227 . . . . . . . . 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 5645 . . . . . . . . 9 ((𝑋 × {𝐵}) “ 𝑋) = ran ((𝑋 × {𝐵}) ↾ 𝑋)
14 ssid 3965 . . . . . . . . . . . . 13 𝑋𝑋
15 xpssres 5973 . . . . . . . . . . . . 13 (𝑋𝑋 → ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵}))
1614, 15ax-mp 5 . . . . . . . . . . . 12 ((𝑋 × {𝐵}) ↾ 𝑋) = (𝑋 × {𝐵})
1716rneqi 5891 . . . . . . . . . . 11 ran ((𝑋 × {𝐵}) ↾ 𝑋) = ran (𝑋 × {𝐵})
18 rnxpss 6123 . . . . . . . . . . 11 ran (𝑋 × {𝐵}) ⊆ {𝐵}
1917, 18eqsstri 3977 . . . . . . . . . 10 ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ {𝐵}
20 simprr 772 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → 𝐵𝑦)
2120snssd 4768 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → {𝐵} ⊆ 𝑦)
2219, 21sstrid 3954 . . . . . . . . 9 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ran ((𝑋 × {𝐵}) ↾ 𝑋) ⊆ 𝑦)
2313, 22eqsstrid 3991 . . . . . . . 8 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)
24 eleq2 2827 . . . . . . . . . 10 (𝑢 = 𝑋 → (𝑥𝑢𝑥𝑋))
25 imaeq2 6008 . . . . . . . . . . 11 (𝑢 = 𝑋 → ((𝑋 × {𝐵}) “ 𝑢) = ((𝑋 × {𝐵}) “ 𝑋))
2625sseq1d 3974 . . . . . . . . . 10 (𝑢 = 𝑋 → (((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦 ↔ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦))
2724, 26anbi12d 632 . . . . . . . . 9 (𝑢 = 𝑋 → ((𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦) ↔ (𝑥𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)))
2827rspcev 3580 . . . . . . . 8 ((𝑋𝐽 ∧ (𝑥𝑋 ∧ ((𝑋 × {𝐵}) “ 𝑋) ⊆ 𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
2911, 12, 23, 28syl12anc 836 . . . . . . 7 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ (𝑦𝐾𝐵𝑦)) → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦))
3029expr 458 . . . . . 6 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (𝐵𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
318, 30sylbid 239 . . . . 5 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) ∧ 𝑦𝐾) → (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
3231ralrimiva 3142 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → ∀𝑦𝐾 (((𝑋 × {𝐵})‘𝑥) ∈ 𝑦 → ∃𝑢𝐽 (𝑥𝑢 ∧ ((𝑋 × {𝐵}) “ 𝑢) ⊆ 𝑦)))
33 simpl1 1192 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝐽 ∈ (TopOn‘𝑋))
34 simpl2 1193 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝐾 ∈ (TopOn‘𝑌))
35 simpr 486 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) ∧ 𝑥𝑋) → 𝑥𝑋)
36 iscnp 22540 . . . . 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 1372 . . . 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 3142 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐵𝑌) → ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))
40 cncnp 22583 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝑋 × {𝐵}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 × {𝐵}):𝑋𝑌 ∧ ∀𝑥𝑋 (𝑋 × {𝐵}) ∈ ((𝐽 CnP 𝐾)‘𝑥))))
41403adant3 1133 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3063  wrex 3072  wss 3909  {csn 4585   × cxp 5630  ran crn 5633  cres 5634  cima 5635  wf 6490  cfv 6494  (class class class)co 7352  TopOnctopon 22211   Cn ccn 22527   CnP ccnp 22528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7665
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-fv 6502  df-ov 7355  df-oprab 7356  df-mpo 7357  df-1st 7914  df-2nd 7915  df-map 8726  df-topgen 17285  df-top 22195  df-topon 22212  df-cn 22530  df-cnp 22531
This theorem is referenced by:  cnconst  22587  xkoccn  22922  txkgen  22955  cnmptc  22965  pcoptcl  24336  blocni  29576  pl1cn  32340  connpconn  33633  cvmliftphtlem  33715  cvmlift3lem9  33725  cnfdmsn  44018  stoweidlem47  44183
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