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Theorem cnmpt2nd 21466
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 7186 . . . . . 6 2nd :V–onto→V
2 fofn 6115 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . . . 5 2nd Fn V
4 ssv 3623 . . . . 5 (𝑋 × 𝑌) ⊆ V
5 fnssres 6002 . . . . 5 ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
63, 4, 5mp2an 708 . . . 4 (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7 dffn5 6239 . . . 4 ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)))
86, 7mpbi 220 . . 3 (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))
9 fvres 6205 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd𝑧))
109mpteq2ia 4738 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧))
11 vex 3201 . . . . 5 𝑥 ∈ V
12 vex 3201 . . . . 5 𝑦 ∈ V
1311, 12op2ndd 7176 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd𝑧) = 𝑦)
1413mpt2mpt 6749 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd𝑧)) = (𝑥𝑋, 𝑦𝑌𝑦)
158, 10, 143eqtri 2647 . 2 (2nd ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝑦)
16 cnmpt21.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
17 cnmpt21.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
18 tx2cn 21407 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
1916, 17, 18syl2anc 693 . 2 (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
2015, 19syl5eqelr 2705 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  Vcvv 3198  wss 3572  cmpt 4727   × cxp 5110  cres 5114   Fn wfn 5881  ontowfo 5884  cfv 5886  (class class class)co 6647  cmpt2 6649  2nd c2nd 7164  TopOnctopon 20709   Cn ccn 21022   ×t ctx 21357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fo 5892  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-map 7856  df-topgen 16098  df-top 20693  df-topon 20710  df-bases 20744  df-cn 21025  df-tx 21359
This theorem is referenced by:  cnmptcom  21475  xkofvcn  21481  cnmptk2  21483  txhmeo  21600  txswaphmeo  21602  ptunhmeo  21605  xkohmeo  21612  tgpsubcn  21888  istgp2  21889  oppgtmd  21895  prdstmdd  21921  dvrcn  21981  divcn  22665  cnrehmeo  22746  htpycom  22769  htpyco1  22771  htpycc  22773  reparphti  22791  pcohtpylem  22813  pcorevlem  22820  cxpcn  24480  vmcn  27538  dipcn  27559  mndpluscn  29957  cvxsconn  31210  cvmlift2lem6  31275
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