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Theorem cnmpt2nd 22276
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 7709 . . . . . 6 2nd :V–onto→V
2 fofn 6591 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . . . 5 2nd Fn V
4 ssv 3990 . . . . 5 (𝑋 × 𝑌) ⊆ V
5 fnssres 6469 . . . . 5 ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
63, 4, 5mp2an 690 . . . 4 (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7 dffn5 6723 . . . 4 ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)))
86, 7mpbi 232 . . 3 (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))
9 fvres 6688 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd𝑧))
109mpteq2ia 5156 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
11 vex 3497 . . . . 5 𝑥 ∈ V
12 vex 3497 . . . . 5 𝑦 ∈ V
1311, 12op2ndd 7699 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
1413mpompt 7265 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
158, 10, 143eqtri 2848 . 2 (2nd ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝑦)
16 cnmpt21.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
17 cnmpt21.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
18 tx2cn 22217 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
1916, 17, 18syl2anc 586 . 2 (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
2015, 19eqeltrrid 2918 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  Vcvv 3494  wss 3935  cmpt 5145   × cxp 5552  cres 5556   Fn wfn 6349  ontowfo 6352  cfv 6354  (class class class)co 7155  cmpo 7157  2nd c2nd 7687  TopOnctopon 21517   Cn ccn 21831   ×t ctx 22167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-map 8407  df-topgen 16716  df-top 21501  df-topon 21518  df-bases 21553  df-cn 21834  df-tx 22169
This theorem is referenced by:  cnmptcom  22285  xkofvcn  22291  cnmptk2  22293  txhmeo  22410  txswaphmeo  22412  ptunhmeo  22415  xkohmeo  22422  tgpsubcn  22697  istgp2  22698  oppgtmd  22704  prdstmdd  22731  dvrcn  22791  divcn  23475  cnrehmeo  23556  htpycom  23579  htpyco1  23581  htpycc  23583  reparphti  23600  pcohtpylem  23622  pcorevlem  23629  cxpcn  25325  vmcn  28475  dipcn  28496  mndpluscn  31169  cvxsconn  32490  cvmlift2lem6  32555
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