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Theorem cnconst2 22686
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 6751 . . 3 (𝐡 ∈ π‘Œ β†’ (𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ)
213ad2ant3 1135 . 2 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) β†’ (𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ)
32adantr 481 . . . 4 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ (𝑋 Γ— {𝐡}):π‘‹βŸΆπ‘Œ)
4 simpll3 1214 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ 𝐡 ∈ π‘Œ)
5 simplr 767 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ π‘₯ ∈ 𝑋)
6 fvconst2g 7171 . . . . . . . 8 ((𝐡 ∈ π‘Œ ∧ π‘₯ ∈ 𝑋) β†’ ((𝑋 Γ— {𝐡})β€˜π‘₯) = 𝐡)
74, 5, 6syl2anc 584 . . . . . . 7 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ 𝐾) β†’ ((𝑋 Γ— {𝐡})β€˜π‘₯) = 𝐡)
87eleq1d 2817 . . . . . 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 1212 . . . . . . . . 9 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
10 toponmax 22327 . . . . . . . . 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 767 . . . . . . . 8 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ π‘₯ ∈ 𝑋)
13 df-ima 5666 . . . . . . . . 9 ((𝑋 Γ— {𝐡}) β€œ 𝑋) = ran ((𝑋 Γ— {𝐡}) β†Ύ 𝑋)
14 ssid 3984 . . . . . . . . . . . . 13 𝑋 βŠ† 𝑋
15 xpssres 5994 . . . . . . . . . . . . 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 5912 . . . . . . . . . . 11 ran ((𝑋 Γ— {𝐡}) β†Ύ 𝑋) = ran (𝑋 Γ— {𝐡})
18 rnxpss 6144 . . . . . . . . . . 11 ran (𝑋 Γ— {𝐡}) βŠ† {𝐡}
1917, 18eqsstri 3996 . . . . . . . . . 10 ran ((𝑋 Γ— {𝐡}) β†Ύ 𝑋) βŠ† {𝐡}
20 simprr 771 . . . . . . . . . . 11 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ 𝐡 ∈ 𝑦)
2120snssd 4789 . . . . . . . . . 10 ((((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) ∧ (𝑦 ∈ 𝐾 ∧ 𝐡 ∈ 𝑦)) β†’ {𝐡} βŠ† 𝑦)
2219, 21sstrid 3973 . . . . . . . . 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 4010 . . . . . . . 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 2821 . . . . . . . . . 10 (𝑒 = 𝑋 β†’ (π‘₯ ∈ 𝑒 ↔ π‘₯ ∈ 𝑋))
25 imaeq2 6029 . . . . . . . . . . 11 (𝑒 = 𝑋 β†’ ((𝑋 Γ— {𝐡}) β€œ 𝑒) = ((𝑋 Γ— {𝐡}) β€œ 𝑋))
2625sseq1d 3993 . . . . . . . . . 10 (𝑒 = 𝑋 β†’ (((𝑋 Γ— {𝐡}) β€œ 𝑒) βŠ† 𝑦 ↔ ((𝑋 Γ— {𝐡}) β€œ 𝑋) βŠ† 𝑦))
2724, 26anbi12d 631 . . . . . . . . 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 3595 . . . . . . . 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 835 . . . . . . 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 457 . . . . . 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 3145 . . . 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 1191 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
34 simpl2 1192 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
35 simpr 485 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ π‘₯ ∈ 𝑋)
36 iscnp 22640 . . . . 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 1371 . . . 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 711 . . 3 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ) ∧ 𝐡 ∈ π‘Œ) ∧ π‘₯ ∈ 𝑋) β†’ (𝑋 Γ— {𝐡}) ∈ ((𝐽 CnP 𝐾)β€˜π‘₯))
3938ralrimiva 3145 . 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 22683 . . 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 1132 . 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 711 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3060  βˆƒwrex 3069   βŠ† wss 3928  {csn 4606   Γ— cxp 5651  ran crn 5654   β†Ύ cres 5655   β€œ cima 5656  βŸΆwf 6512  β€˜cfv 6516  (class class class)co 7377  TopOnctopon 22311   Cn ccn 22627   CnP ccnp 22628
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-fv 6524  df-ov 7380  df-oprab 7381  df-mpo 7382  df-1st 7941  df-2nd 7942  df-map 8789  df-topgen 17354  df-top 22295  df-topon 22312  df-cn 22630  df-cnp 22631
This theorem is referenced by:  cnconst  22687  xkoccn  23022  txkgen  23055  cnmptc  23065  pcoptcl  24436  blocni  29844  pl1cn  32659  connpconn  33950  cvmliftphtlem  34032  cvmlift3lem9  34042  cnfdmsn  44276  stoweidlem47  44441
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