MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnmpt2nd Structured version   Visualization version   GIF version

Theorem cnmpt2nd 23560
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 8008 . . . . . 6 2nd :V–onto→V
2 fofn 6807 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . . . 5 2nd Fn V
4 ssv 4002 . . . . 5 (𝑋 × 𝑌) ⊆ V
5 fnssres 6672 . . . . 5 ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
63, 4, 5mp2an 691 . . . 4 (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7 dffn5 6951 . . . 4 ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)))
86, 7mpbi 229 . . 3 (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))
9 fvres 6910 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd𝑧))
109mpteq2ia 5245 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
11 vex 3473 . . . . 5 𝑥 ∈ V
12 vex 3473 . . . . 5 𝑦 ∈ V
1311, 12op2ndd 7998 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
1413mpompt 7528 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
158, 10, 143eqtri 2759 . 2 (2nd ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝑦)
16 cnmpt21.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
17 cnmpt21.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
18 tx2cn 23501 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
1916, 17, 18syl2anc 583 . 2 (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
2015, 19eqeltrrid 2833 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  Vcvv 3469  wss 3944  cmpt 5225   × cxp 5670  cres 5674   Fn wfn 6537  ontowfo 6540  cfv 6542  (class class class)co 7414  cmpo 7416  2nd c2nd 7986  TopOnctopon 22799   Cn ccn 23115   ×t ctx 23451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8838  df-topgen 17416  df-top 22783  df-topon 22800  df-bases 22836  df-cn 23118  df-tx 23453
This theorem is referenced by:  cnmptcom  23569  xkofvcn  23575  cnmptk2  23577  txhmeo  23694  txswaphmeo  23696  ptunhmeo  23699  xkohmeo  23706  tgpsubcn  23981  istgp2  23982  oppgtmd  23988  prdstmdd  24015  dvrcn  24075  divcnOLD  24771  divcn  24773  cnrehmeo  24865  cnrehmeoOLD  24866  htpycom  24889  htpyco1  24891  htpycc  24893  reparphti  24910  reparphtiOLD  24911  pcohtpylem  24933  pcorevlem  24940  cxpcn  26666  cxpcnOLD  26667  vmcn  30496  dipcn  30517  mndpluscn  33463  cvxsconn  34789  cvmlift2lem6  34854
  Copyright terms: Public domain W3C validator