Theorem cnmpt1st 14456

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	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))

Assertion

Ref	Expression
cnmpt1st	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((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.

Step	Hyp	Ref	Expression
1		fo1st 6210	. . . . . 6 ⊢ 1st :V–onto→V
2		fofn 5478	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1st Fn V
4		ssv 3201	. . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V
5		fnssres 5367	. . . . 5 ⊢ ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6	3, 4, 5	mp2an 426	. . . 4 ⊢ (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7		dffn5im 5602	. . . 4 ⊢ ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) → (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)))
8	6, 7	ax-mp 5	. . 3 ⊢ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧))
9		fvres 5578	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st ‘𝑧))
10	9	mpteq2ia 4115	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧))
11		vex 2763	. . . . 5 ⊢ 𝑥 ∈ V
12		vex 2763	. . . . 5 ⊢ 𝑦 ∈ V
13	11, 12	op1std 6201	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
14	13	mpompt 6010	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥)
15	8, 10, 14	3eqtri 2218	. 2 ⊢ (1st ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥)
16		cnmpt21.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
17		cnmpt21.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
18		tx1cn 14437	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
19	16, 17, 18	syl2anc 411	. 2 ⊢ (𝜑 → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
20	15, 19	eqeltrrid 2281	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3153   ↦ cmpt 4090   × cxp 4657   ↾ cres 4661   Fn wfn 5249  –onto→wfo 5252  ‘cfv 5254  (class class class)co 5918   ∈ cmpo 5920  1^st c1st 6191  TopOnctopon 14178   Cn ccn 14353   ×_t ctx 14420

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-map 6704  df-topgen 12871  df-top 14166  df-topon 14179  df-bases 14211  df-cn 14356  df-tx 14421

This theorem is referenced by:  cnmptcom  14466  txhmeo  14487  txswaphmeo  14489  divcnap  14723  cnrehmeocntop  14764