Theorem cnmpt1st 23593

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 7950	. . . . . 6 ⊢ 1st :V–onto→V
2		fofn 6745	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1st Fn V
4		ssv 3956	. . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V
5		fnssres 6612	. . . . 5 ⊢ ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6	3, 4, 5	mp2an 692	. . . 4 ⊢ (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7		dffn5 6889	. . . 4 ⊢ ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)))
8	6, 7	mpbi 230	. . 3 ⊢ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧))
9		fvres 6850	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st ‘𝑧))
10	9	mpteq2ia 5190	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧))
11		vex 3442	. . . . 5 ⊢ 𝑥 ∈ V
12		vex 3442	. . . . 5 ⊢ 𝑦 ∈ V
13	11, 12	op1std 7940	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
14	13	mpompt 7469	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥)
15	8, 10, 14	3eqtri 2760	. 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 23534	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
19	16, 17, 18	syl2anc 584	. 2 ⊢ (𝜑 → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
20	15, 19	eqeltrrid 2838	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  Vcvv 3438   ⊆ wss 3899   ↦ cmpt 5176   × cxp 5619   ↾ cres 5623   Fn wfn 6484  –onto→wfo 6487  ‘cfv 6489  (class class class)co 7355   ∈ cmpo 7357  1^st c1st 7928  TopOnctopon 22835   Cn ccn 23149   ×_t ctx 23485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fo 6495  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-map 8761  df-topgen 17357  df-top 22819  df-topon 22836  df-bases 22871  df-cn 23152  df-tx 23487

This theorem is referenced by:  cnmptcom  23603  xkofvcn  23609  cnmptk2  23611  txhmeo  23728  txswaphmeo  23730  ptunhmeo  23733  xkohmeo  23740  tgpsubcn  24015  istgp2  24016  oppgtmd  24022  prdstmdd  24049  dvrcn  24109  divcnOLD  24794  divcn  24796  cnrehmeo  24888  cnrehmeoOLD  24889  htpycom  24912  htpyid  24913  htpyco1  24914  htpycc  24916  reparphti  24933  reparphtiOLD  24934  pcocn  24954  pcohtpylem  24956  pcopt  24959  pcopt2  24960  pcoass  24961  pcorevlem  24963  cxpcn  26691  cxpcnOLD  26692  vmcn  30690  dipcn  30711  mndpluscn  33950  cvxsconn  35298  cvmlift2lem12  35369