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Theorem cnmpt1st 22204
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 (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k (𝜑𝐾 ∈ (TopOn‘𝑌))
Assertion
Ref Expression
cnmpt1st (𝜑 → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
Distinct variable groups:   𝑥,𝑦,𝜑   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦
Allowed substitution hints:   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt1st
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fo1st 7698 . . . . . 6 1st :V–onto→V
2 fofn 6585 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 ssv 3988 . . . . 5 (𝑋 × 𝑌) ⊆ V
5 fnssres 6463 . . . . 5 ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
63, 4, 5mp2an 688 . . . 4 (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7 dffn5 6717 . . . 4 ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)))
86, 7mpbi 231 . . 3 (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧))
9 fvres 6682 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st𝑧))
109mpteq2ia 5148 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧))
11 vex 3495 . . . . 5 𝑥 ∈ V
12 vex 3495 . . . . 5 𝑦 ∈ V
1311, 12op1std 7688 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
1413mpompt 7255 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
158, 10, 143eqtri 2845 . 2 (1st ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝑥)
16 cnmpt21.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
17 cnmpt21.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
18 tx1cn 22145 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
1916, 17, 18syl2anc 584 . 2 (𝜑 → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
2015, 19eqeltrrid 2915 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  cmpt 5137   × cxp 5546  cres 5550   Fn wfn 6343  ontowfo 6346  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  TopOnctopon 21446   Cn ccn 21760   ×t ctx 22096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763  df-tx 22098
This theorem is referenced by:  cnmptcom  22214  xkofvcn  22220  cnmptk2  22222  txhmeo  22339  txswaphmeo  22341  ptunhmeo  22344  xkohmeo  22351  tgpsubcn  22626  istgp2  22627  oppgtmd  22633  prdstmdd  22659  dvrcn  22719  divcn  23403  cnrehmeo  23484  htpycom  23507  htpyid  23508  htpyco1  23509  htpycc  23511  reparphti  23528  pcocn  23548  pcohtpylem  23550  pcopt  23553  pcopt2  23554  pcoass  23555  pcorevlem  23557  cxpcn  25253  vmcn  28403  dipcn  28424  mndpluscn  31068  cvxsconn  32387  cvmlift2lem12  32458
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