Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnmpt11f GIF version

Theorem cnmpt11f 12526
 Description: The composition of continuous functions is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptid.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt11.a (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
cnmpt11f.f (𝜑𝐹 ∈ (𝐾 Cn 𝐿))
Assertion
Ref Expression
cnmpt11f (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))
Distinct variable groups:   𝑥,𝐹   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋   𝑥,𝐾   𝑥,𝐿
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem cnmpt11f
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 cnmptid.j . 2 (𝜑𝐽 ∈ (TopOn‘𝑋))
2 cnmpt11.a . 2 (𝜑 → (𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾))
3 cntop2 12444 . . . 4 ((𝑥𝑋𝐴) ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
42, 3syl 14 . . 3 (𝜑𝐾 ∈ Top)
5 eqid 2140 . . . 4 𝐾 = 𝐾
65toptopon 12258 . . 3 (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘ 𝐾))
74, 6sylib 121 . 2 (𝜑𝐾 ∈ (TopOn‘ 𝐾))
8 cnmpt11f.f . . . . 5 (𝜑𝐹 ∈ (𝐾 Cn 𝐿))
9 eqid 2140 . . . . . 6 𝐿 = 𝐿
105, 9cnf 12446 . . . . 5 (𝐹 ∈ (𝐾 Cn 𝐿) → 𝐹: 𝐾 𝐿)
118, 10syl 14 . . . 4 (𝜑𝐹: 𝐾 𝐿)
1211feqmptd 5485 . . 3 (𝜑𝐹 = (𝑦 𝐾 ↦ (𝐹𝑦)))
1312, 8eqeltrrd 2218 . 2 (𝜑 → (𝑦 𝐾 ↦ (𝐹𝑦)) ∈ (𝐾 Cn 𝐿))
14 fveq2 5432 . 2 (𝑦 = 𝐴 → (𝐹𝑦) = (𝐹𝐴))
151, 2, 7, 13, 14cnmpt11 12525 1 (𝜑 → (𝑥𝑋 ↦ (𝐹𝐴)) ∈ (𝐽 Cn 𝐿))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1481  ∪ cuni 3745   ↦ cmpt 3998  ⟶wf 5130  ‘cfv 5134  (class class class)co 5785  Topctop 12237  TopOnctopon 12250   Cn ccn 12427 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-sep 4055  ax-pow 4107  ax-pr 4141  ax-un 4365  ax-setind 4462 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-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 4225  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-rn 4561  df-res 4562  df-ima 4563  df-iota 5099  df-fun 5136  df-fn 5137  df-f 5138  df-fv 5142  df-ov 5788  df-oprab 5789  df-mpo 5790  df-1st 6049  df-2nd 6050  df-map 6555  df-top 12238  df-topon 12251  df-cn 12430 This theorem is referenced by:  cnmpt12f  12528
 Copyright terms: Public domain W3C validator