Theorem cnmpt1st 23716

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 7985	. . . . . 6 ⊢ 1st :V–onto→V
2		fofn 6775	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1st Fn V
4		ssv 3958	. . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V
5		fnssres 6639	. . . . 5 ⊢ ((1st Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6	3, 4, 5	mp2an 702	. . . 4 ⊢ (1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7		dffn5 6920	. . . 4 ⊢ ((1st ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)))
8	6, 7	mpbi 232	. . 3 ⊢ (1st ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧))
9		fvres 6881	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((1st ↾ (𝑋 × 𝑌))‘𝑧) = (1st ‘𝑧))
10	9	mpteq2ia 5192	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((1st ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧))
11		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
12		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
13	11, 12	op1std 7975	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑧) = 𝑥)
14	13	mpompt 7505	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (1st ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥)
15	8, 10, 14	3eqtri 2788	. 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 23657	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
19	16, 17, 18	syl2anc 593	. 2 ⊢ (𝜑 → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))
20	15, 19	eqeltrrid 2866	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑥) ∈ ((𝐽 ×t 𝐾) Cn 𝐽))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3902   ↦ cmpt 5178   × cxp 5641   ↾ cres 5645   Fn wfn 6511  –onto→wfo 6514  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  1^st c1st 7963  TopOnctopon 22958   Cn ccn 23272   ×_t ctx 23608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fo 6522  df-fv 6524  df-ov 7394  df-oprab 7395  df-mpo 7396  df-1st 7965  df-2nd 7966  df-map 8804  df-topgen 17463  df-top 22942  df-topon 22959  df-bases 22994  df-cn 23275  df-tx 23610

This theorem is referenced by:  cnmptcom  23726  xkofvcn  23732  cnmptk2  23734  txhmeo  23851  txswaphmeo  23853  ptunhmeo  23856  xkohmeo  23863  tgpsubcn  24138  istgp2  24139  oppgtmd  24145  prdstmdd  24172  dvrcn  24232  divcn  24918  cnrehmeo  25003  htpycom  25026  htpyid  25027  htpyco1  25028  htpycc  25030  reparphti  25047  pcocn  25067  pcohtpylem  25069  pcopt  25072  pcopt2  25073  pcoass  25074  pcorevlem  25076  cxpcn  26798  vmcn  30859  dipcn  30880  mndpluscn  34184  cvxsconn  35554  cvmlift2lem12  35625