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Theorem cnmpt2nd 12467
Description: The projection onto the second coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
Assertion
Ref Expression
cnmpt2nd (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
Distinct variable groups:   𝑥,𝑦,𝜑   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2nd
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fo2nd 6056 . . . . . 6 2nd :V–onto→V
2 fofn 5347 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . . . 5 2nd Fn V
4 ssv 3119 . . . . 5 (𝑋 × 𝑌) ⊆ V
5 fnssres 5236 . . . . 5 ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
63, 4, 5mp2an 422 . . . 4 (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7 dffn5im 5467 . . . 4 ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)))
86, 7ax-mp 5 . . 3 (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))
9 fvres 5445 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd𝑧))
109mpteq2ia 4014 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
11 vex 2689 . . . . 5 𝑥 ∈ V
12 vex 2689 . . . . 5 𝑦 ∈ V
1311, 12op2ndd 6047 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
1413mpompt 5863 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
158, 10, 143eqtri 2164 . 2 (2nd ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝑦)
16 cnmpt21.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
17 cnmpt21.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
18 tx2cn 12448 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
1916, 17, 18syl2anc 408 . 2 (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
2015, 19eqeltrrid 2227 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  Vcvv 2686  wss 3071  cmpt 3989   × cxp 4537  cres 4541   Fn wfn 5118  ontowfo 5121  cfv 5123  (class class class)co 5774  cmpo 5776  2nd c2nd 6037  TopOnctopon 12186   Cn ccn 12363   ×t ctx 12430
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-topgen 12150  df-top 12174  df-topon 12187  df-bases 12219  df-cn 12366  df-tx 12431
This theorem is referenced by:  cnmptcom  12476  txhmeo  12497  txswaphmeo  12499  divcnap  12733  cnrehmeocntop  12771
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