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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 6806 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 ssv 4005 . . . . 5 (𝑋 Γ— π‘Œ) βŠ† V
5 fnssres 6672 . . . . 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 688 . . . 4 (1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ)
7 dffn5 6949 . . . 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 6909 . . . 4 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
109mpteq2ia 5250 . . 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 3476 . . . . 5 π‘₯ ∈ V
12 vex 3476 . . . . 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 2762 . 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 582 . 2 (πœ‘ β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
2015, 19eqeltrrid 2836 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 1539   ∈ wcel 2104  Vcvv 3472   βŠ† wss 3947   ↦ cmpt 5230   Γ— cxp 5673   β†Ύ cres 5677   Fn wfn 6537  β€“ontoβ†’wfo 6540  β€˜cfv 6542  (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 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  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  24698  cnrehmeoOLD  24699  htpycom  24722  htpyid  24723  htpyco1  24724  htpycc  24726  reparphti  24743  reparphtiOLD  24744  pcocn  24764  pcohtpylem  24766  pcopt  24769  pcopt2  24770  pcoass  24771  pcorevlem  24773  cxpcn  26489  vmcn  30219  dipcn  30240  mndpluscn  33204  cvxsconn  34532  cvmlift2lem12  34603  gg-cxpcn  35470
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