Theorem cnmpt2nd 23698

Description: The projection onto the second 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
cnmpt2nd	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))

Distinct variable groups:   𝑥,𝑦,𝜑   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐽(𝑥,𝑦)   𝐾(𝑥,𝑦)

Proof of Theorem cnmpt2nd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo2nd 8051	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6836	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		ssv 4033	. . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V
5		fnssres 6703	. . . . 5 ⊢ ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6	3, 4, 5	mp2an 691	. . . 4 ⊢ (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7		dffn5 6980	. . . 4 ⊢ ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)))
8	6, 7	mpbi 230	. . 3 ⊢ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))
9		fvres 6939	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd ‘𝑧))
10	9	mpteq2ia 5269	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧))
11		vex 3492	. . . . 5 ⊢ 𝑥 ∈ V
12		vex 3492	. . . . 5 ⊢ 𝑦 ∈ V
13	11, 12	op2ndd 8041	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
14	13	mpompt 7564	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦)
15	8, 10, 14	3eqtri 2772	. 2 ⊢ (2nd ↾ (𝑋 × 𝑌)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦)
16		cnmpt21.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
17		cnmpt21.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
18		tx2cn 23639	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
19	16, 17, 18	syl2anc 583	. 2 ⊢ (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
20	15, 19	eqeltrrid 2849	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976   ↦ cmpt 5249   × cxp 5698   ↾ cres 5702   Fn wfn 6568  –onto→wfo 6571  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  2^nd c2nd 8029  TopOnctopon 22937   Cn ccn 23253   ×_t ctx 23589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-map 8886  df-topgen 17503  df-top 22921  df-topon 22938  df-bases 22974  df-cn 23256  df-tx 23591

This theorem is referenced by:  cnmptcom  23707  xkofvcn  23713  cnmptk2  23715  txhmeo  23832  txswaphmeo  23834  ptunhmeo  23837  xkohmeo  23844  tgpsubcn  24119  istgp2  24120  oppgtmd  24126  prdstmdd  24153  dvrcn  24213  divcnOLD  24909  divcn  24911  cnrehmeo  25003  cnrehmeoOLD  25004  htpycom  25027  htpyco1  25029  htpycc  25031  reparphti  25048  reparphtiOLD  25049  pcohtpylem  25071  pcorevlem  25078  cxpcn  26805  cxpcnOLD  26806  vmcn  30731  dipcn  30752  mndpluscn  33872  cvxsconn  35211  cvmlift2lem6  35276