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Theorem cnmpt1st 13873
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 6160 . . . . . 6 1st :V–ontoβ†’V
2 fofn 5442 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 ssv 3179 . . . . 5 (𝑋 Γ— π‘Œ) βŠ† V
5 fnssres 5331 . . . . 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 426 . . . 4 (1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ)
7 dffn5im 5563 . . . 4 ((1st β†Ύ (𝑋 Γ— π‘Œ)) Fn (𝑋 Γ— π‘Œ) β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) = (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§)))
86, 7ax-mp 5 . . 3 (1st β†Ύ (𝑋 Γ— π‘Œ)) = (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§))
9 fvres 5541 . . . 4 (𝑧 ∈ (𝑋 Γ— π‘Œ) β†’ ((1st β†Ύ (𝑋 Γ— π‘Œ))β€˜π‘§) = (1st β€˜π‘§))
109mpteq2ia 4091 . . 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 2742 . . . . 5 π‘₯ ∈ V
12 vex 2742 . . . . 5 𝑦 ∈ V
1311, 12op1std 6151 . . . 4 (𝑧 = ⟨π‘₯, π‘¦βŸ© β†’ (1st β€˜π‘§) = π‘₯)
1413mpompt 5969 . . 3 (𝑧 ∈ (𝑋 Γ— π‘Œ) ↦ (1st β€˜π‘§)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯)
158, 10, 143eqtri 2202 . 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 13854 . . 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 411 . 2 (πœ‘ β†’ (1st β†Ύ (𝑋 Γ— π‘Œ)) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
2015, 19eqeltrrid 2265 1 (πœ‘ β†’ (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ π‘₯) ∈ ((𝐽 Γ—t 𝐾) Cn 𝐽))
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2739   βŠ† wss 3131   ↦ cmpt 4066   Γ— cxp 4626   β†Ύ cres 4630   Fn wfn 5213  β€“ontoβ†’wfo 5216  β€˜cfv 5218  (class class class)co 5877   ∈ cmpo 5879  1st c1st 6141  TopOnctopon 13595   Cn ccn 13770   Γ—t ctx 13837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-map 6652  df-topgen 12714  df-top 13583  df-topon 13596  df-bases 13628  df-cn 13773  df-tx 13838
This theorem is referenced by:  cnmptcom  13883  txhmeo  13904  txswaphmeo  13906  divcnap  14140  cnrehmeocntop  14178
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