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Theorem cnmpt2c 12518
 Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmpt2c.l (𝜑𝐿 ∈ (TopOn‘𝑍))
cnmpt2c.p (𝜑𝑃𝑍)
Assertion
Ref Expression
cnmpt2c (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑃,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦
Allowed substitution hints:   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2c
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqidd 2141 . . 3 (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑃 = 𝑃)
21mpompt 5872 . 2 (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥𝑋, 𝑦𝑌𝑃)
3 cnmpt21.j . . . 4 (𝜑𝐽 ∈ (TopOn‘𝑋))
4 cnmpt21.k . . . 4 (𝜑𝐾 ∈ (TopOn‘𝑌))
5 txtopon 12490 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
63, 4, 5syl2anc 409 . . 3 (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7 cnmpt2c.l . . 3 (𝜑𝐿 ∈ (TopOn‘𝑍))
8 cnmpt2c.p . . 3 (𝜑𝑃𝑍)
96, 7, 8cnmptc 12510 . 2 (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
102, 9eqeltrrid 2228 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  ⟨cop 3536   ↦ cmpt 3998   × cxp 4546  ‘cfv 5132  (class class class)co 5783   ∈ cmpo 5785  TopOnctopon 12236   Cn ccn 12413   ×t ctx 12480 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-map 6553  df-topgen 12200  df-top 12224  df-topon 12237  df-bases 12269  df-cn 12416  df-cnp 12417  df-tx 12481 This theorem is referenced by:  cnrehmeocntop  12821
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