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Theorem cnmpt1st 23035
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 7942 . . . . . 6 1st :V–ontoβ†’V
2 fofn 6759 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 ssv 3969 . . . . 5 (𝑋 Γ— π‘Œ) βŠ† V
5 fnssres 6625 . . . . 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 691 . . . 4 (1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ)
7 dffn5 6902 . . . 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 6862 . . . 4 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
109mpteq2ia 5209 . . 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 3448 . . . . 5 π‘₯ ∈ V
12 vex 3448 . . . . 5 𝑦 ∈ V
1311, 12op1std 7932 . . . 4 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ (1st β€˜π‘§) = π‘₯)
1413mpompt 7471 . . 3 (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ (1st β€˜π‘§)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯)
158, 10, 143eqtri 2765 . 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 22976 . . 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 585 . 2 (πœ‘ β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
2015, 19eqeltrrid 2839 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 1542   ∈ wcel 2107  Vcvv 3444   βŠ† wss 3911   ↦ cmpt 5189   Γ— cxp 5632   β†Ύ cres 5636   Fn wfn 6492  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  TopOnctopon 22275   Cn ccn 22591   Γ—t ctx 22927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-topgen 17330  df-top 22259  df-topon 22276  df-bases 22312  df-cn 22594  df-tx 22929
This theorem is referenced by:  cnmptcom  23045  xkofvcn  23051  cnmptk2  23053  txhmeo  23170  txswaphmeo  23172  ptunhmeo  23175  xkohmeo  23182  tgpsubcn  23457  istgp2  23458  oppgtmd  23464  prdstmdd  23491  dvrcn  23551  divcn  24247  cnrehmeo  24332  htpycom  24355  htpyid  24356  htpyco1  24357  htpycc  24359  reparphti  24376  pcocn  24396  pcohtpylem  24398  pcopt  24401  pcopt2  24402  pcoass  24403  pcorevlem  24405  cxpcn  26114  vmcn  29683  dipcn  29704  mndpluscn  32564  cvxsconn  33894  cvmlift2lem12  33965
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