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Theorem cnconst2 23007
Description: A constant function is continuous. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
cnconst2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) β†’ (𝑋 Γ— {𝐡}) ∈ (𝐽 Cn 𝐾))
Proof of Theorem cnconst2
Dummy variables π‘₯ 𝑒 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fconst6g 6779 . . 3 (𝐡 ∈ π‘Œ β†’ (𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ)
213ad2ant3 1133 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) β†’ (𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ)
32adantr 479 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ (𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ)
4 simpll3 1212 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ 𝐡 ∈ π‘Œ)
5 simplr 765 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ π‘₯ ∈ 𝑋)
6 fvconst2g 7204 . . . . . . . 8 ((𝐡 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑋 Γ— {𝐡})β€˜π‘₯) = 𝐡)
74, 5, 6syl2anc 582 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ ((𝑋 Γ— {𝐡})β€˜π‘₯) = 𝐡)
87eleq1d 2816 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ (((𝑋 Γ— {𝐡})β€˜π‘₯) ∈ 𝑦 ↔ 𝐡 ∈ 𝑦))
9 simpll1 1210 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
10 toponmax 22648 . . . . . . . . 9 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
119, 10syl 17 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ 𝑋 ∈ 𝐽)
12 simplr 765 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ π‘₯ ∈ 𝑋)
13 df-ima 5688 . . . . . . . . 9 ((𝑋 Γ— {𝐡}) β€œ 𝑋) = ran ((𝑋 Γ— {𝐡}) β†Ύ 𝑋)
14 ssid 4003 . . . . . . . . . . . . 13 𝑋 βŠ† 𝑋
15 xpssres 6017 . . . . . . . . . . . . 13 (𝑋 βŠ† 𝑋 β†’ ((𝑋 Γ— {𝐡}) β†Ύ 𝑋) = (𝑋 Γ— {𝐡}))
1614, 15ax-mp 5 . . . . . . . . . . . 12 ((𝑋 Γ— {𝐡}) β†Ύ 𝑋) = (𝑋 Γ— {𝐡})
1716rneqi 5935 . . . . . . . . . . 11 ran ((𝑋 Γ— {𝐡}) β†Ύ 𝑋) = ran (𝑋 Γ— {𝐡})
18 rnxpss 6170 . . . . . . . . . . 11 ran (𝑋 Γ— {𝐡}) βŠ† {𝐡}
1917, 18eqsstri 4015 . . . . . . . . . 10 ran ((𝑋 Γ— {𝐡}) β†Ύ 𝑋) βŠ† {𝐡}
20 simprr 769 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ 𝐡 ∈ 𝑦)
2120snssd 4811 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ {𝐡} βŠ† 𝑦)
2219, 21sstrid 3992 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ ran ((𝑋 Γ— {𝐡}) β†Ύ 𝑋) βŠ† 𝑦)
2313, 22eqsstrid 4029 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ ((𝑋 Γ— {𝐡}) β€œ 𝑋) βŠ† 𝑦)
24 eleq2 2820 . . . . . . . . . 10 (𝑒 = 𝑋 β†’ (π‘₯ ∈ 𝑒 ↔ π‘₯ ∈ 𝑋))
25 imaeq2 6054 . . . . . . . . . . 11 (𝑒 = 𝑋 β†’ ((𝑋 Γ— {𝐡}) β€œ 𝑒) = ((𝑋 Γ— {𝐡}) β€œ 𝑋))
2625sseq1d 4012 . . . . . . . . . 10 (𝑒 = 𝑋 β†’ (((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦 ↔ ((𝑋 Γ— {𝐡}) β€œ 𝑋) βŠ† 𝑦))
2724, 26anbi12d 629 . . . . . . . . 9 (𝑒 = 𝑋 β†’ ((π‘₯ ∈ 𝑒 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦) ↔ (π‘₯ ∈ 𝑋 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑋) βŠ† 𝑦)))
2827rspcev 3611 . . . . . . . 8 ((𝑋 ∈ 𝐽 ∧ (π‘₯ ∈ 𝑋 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑋) βŠ† 𝑦)) β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦))
2911, 12, 23, 28syl12anc 833 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦))
3029expr 455 . . . . . 6 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ (𝐡 ∈ 𝑦 β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦)))
318, 30sylbid 239 . . . . 5 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ (((𝑋 Γ— {𝐡})β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦)))
3231ralrimiva 3144 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ βˆ€π‘¦ ∈ 𝐾 (((𝑋 Γ— {𝐡})β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦)))
33 simpl1 1189 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
34 simpl2 1190 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
35 simpr 483 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
36 iscnp 22961 . . . . 5 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ ((𝑋 Γ— {𝐡}) ∈ ((𝐽 CnP 𝐾)β€˜π‘₯) ↔ ((𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 (((𝑋 Γ— {𝐡})β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦)))))
3733, 34, 35, 36syl3anc 1369 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ ((𝑋 Γ— {𝐡}) ∈ ((𝐽 CnP 𝐾)β€˜π‘₯) ↔ ((𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ ∧ βˆ€π‘¦ ∈ 𝐾 (((𝑋 Γ— {𝐡})β€˜π‘₯) ∈ 𝑦 β†’ βˆƒπ‘’ ∈ 𝐽 (π‘₯ ∈ 𝑒 ∧ ((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦)))))
383, 32, 37mpbir2and 709 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ (𝑋 Γ— {𝐡}) ∈ ((𝐽 CnP 𝐾)β€˜π‘₯))
3938ralrimiva 3144 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) β†’ βˆ€π‘₯ ∈ 𝑋 (𝑋 Γ— {𝐡}) ∈ ((𝐽 CnP 𝐾)β€˜π‘₯))
40 cncnp 23004 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ ((𝑋 Γ— {𝐡}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 (𝑋 Γ— {𝐡}) ∈ ((𝐽 CnP 𝐾)β€˜π‘₯))))
41403adant3 1130 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) β†’ ((𝑋 Γ— {𝐡}) ∈ (𝐽 Cn 𝐾) ↔ ((𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 (𝑋 Γ— {𝐡}) ∈ ((𝐽 CnP 𝐾)β€˜π‘₯))))
422, 39, 41mpbir2and 709 1 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) β†’ (𝑋 Γ— {𝐡}) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   βŠ† wss 3947  {csn 4627   Γ— cxp 5673  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7411  TopOnctopon 22632   Cn ccn 22948   CnP ccnp 22949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-topgen 17393  df-top 22616  df-topon 22633  df-cn 22951  df-cnp 22952
This theorem is referenced by:  cnconst  23008  xkoccn  23343  txkgen  23376  cnmptc  23386  pcoptcl  24768  blocni  30325  pl1cn  33233  connpconn  34524  cvmliftphtlem  34606  cvmlift3lem9  34616  cnfdmsn  44896  stoweidlem47  45061
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