ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  cnmpt1st GIF version

Theorem cnmpt1st 13082
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 6136 . . . . . 6 1st :V–onto→V
2 fofn 5422 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 ssv 3169 . . . . 5 (𝑋 × 𝑌) ⊆ V
5 fnssres 5311 . . . . 5 ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
63, 4, 5mp2an 424 . . . 4 (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7 dffn5im 5542 . . . 4 ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)))
86, 7ax-mp 5 . . 3 (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧))
9 fvres 5520 . . . 4 (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st𝑧))
109mpteq2ia 4075 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧))
11 vex 2733 . . . . 5 𝑥 ∈ V
12 vex 2733 . . . . 5 𝑦 ∈ V
1311, 12op1std 6127 . . . 4 (𝑧 = ⟨𝑥, 𝑦⟩ → (1st𝑧) = 𝑥)
1413mpompt 5945 . . 3 (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st𝑧)) = (𝑥𝑋, 𝑦𝑌𝑥)
158, 10, 143eqtri 2195 . 2 (1st ↾ (𝑋 × 𝑌)) = (𝑥𝑋, 𝑦𝑌𝑥)
16 cnmpt21.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
17 cnmpt21.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
18 tx1cn 13063 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
1916, 17, 18syl2anc 409 . 2 (𝜑 → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
2015, 19eqeltrrid 2258 1 (𝜑 → (𝑥𝑋, 𝑦𝑌𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wcel 2141  Vcvv 2730  wss 3121  cmpt 4050   × cxp 4609  cres 4613   Fn wfn 5193  ontowfo 5196  cfv 5198  (class class class)co 5853  cmpo 5855  1st c1st 6117  TopOnctopon 12802   Cn ccn 12979   ×t ctx 13046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-topgen 12600  df-top 12790  df-topon 12803  df-bases 12835  df-cn 12982  df-tx 13047
This theorem is referenced by:  cnmptcom  13092  txhmeo  13113  txswaphmeo  13115  divcnap  13349  cnrehmeocntop  13387
  Copyright terms: Public domain W3C validator