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Theorem cnmpt1st 23392
Description: The projection onto the first coordinate is continuous. (Contributed by Mario Carneiro, 6-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmpt21.j (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
cnmpt21.k (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
Assertion
Ref Expression
cnmpt1st (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
Distinct variable groups:   π‘₯,𝑦,πœ‘   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   𝐽(π‘₯,𝑦)   𝐾(π‘₯,𝑦)
Proof of Theorem cnmpt1st
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fo1st 7997 . . . . . 6 1st :V–ontoβ†’V
2 fofn 6807 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 ssv 4006 . . . . 5 (𝑋 Γ— π‘Œ) βŠ† V
5 fnssres 6673 . . . . 5 ((1st Fn V ∧ (𝑋 Γ— π‘Œ) βŠ† V) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ))
63, 4, 5mp2an 690 . . . 4 (1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ)
7 dffn5 6950 . . . 4 ((1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ) ↔ (1st β†Ύ (𝑋 Γ— π‘Œ)) = (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§)))
86, 7mpbi 229 . . 3 (1st β†Ύ (𝑋 Γ— π‘Œ)) = (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§))
9 fvres 6910 . . . 4 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
109mpteq2ia 5251 . . 3 (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§)) = (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ (1st β€˜π‘§))
11 vex 3478 . . . . 5 π‘₯ ∈ V
12 vex 3478 . . . . 5 𝑦 ∈ V
1311, 12op1std 7987 . . . 4 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ (1st β€˜π‘§) = π‘₯)
1413mpompt 7524 . . 3 (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ (1st β€˜π‘§)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯)
158, 10, 143eqtri 2764 . 2 (1st β†Ύ (𝑋 Γ— π‘Œ)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯)
16 cnmpt21.j . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
17 cnmpt21.k . . 3 (πœ‘ β†’ 𝐾 ∈ (TopOnβ€˜π‘Œ))
18 tx1cn 23333 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ 𝐾 ∈ (TopOnβ€˜π‘Œ)) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
1916, 17, 18syl2anc 584 . 2 (πœ‘ β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
2015, 19eqeltrrid 2838 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  Vcvv 3474   βŠ† wss 3948   ↦ cmpt 5231   Γ— cxp 5674   β†Ύ cres 5678   Fn wfn 6538  β€“ontoβ†’wfo 6541  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413  1st c1st 7975  TopOnctopon 22632   Cn ccn 22948   Γ—t ctx 23284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-map 8824  df-topgen 17393  df-top 22616  df-topon 22633  df-bases 22669  df-cn 22951  df-tx 23286
This theorem is referenced by:  cnmptcom  23402  xkofvcn  23408  cnmptk2  23410  txhmeo  23527  txswaphmeo  23529  ptunhmeo  23532  xkohmeo  23539  tgpsubcn  23814  istgp2  23815  oppgtmd  23821  prdstmdd  23848  dvrcn  23908  divcnOLD  24604  divcn  24606  cnrehmeo  24693  htpycom  24716  htpyid  24717  htpyco1  24718  htpycc  24720  reparphti  24737  pcocn  24757  pcohtpylem  24759  pcopt  24762  pcopt2  24763  pcoass  24764  pcorevlem  24766  cxpcn  26477  vmcn  30207  dipcn  30228  mndpluscn  33192  cvxsconn  34520  cvmlift2lem12  34591  gg-cnrehmeo  35457  gg-reparphti  35458  gg-cxpcn  35470
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