Theorem cnmpt2nd 23901

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 8011	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6795	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		ssv 3958	. . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V
5		fnssres 6659	. . . . 5 ⊢ ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6	3, 4, 5	mp2an 705	. . . 4 ⊢ (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7		dffn5 6940	. . . 4 ⊢ ((2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌) ↔ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)))
8	6, 7	mpbi 233	. . 3 ⊢ (2nd ↾ (𝑋 × 𝑌)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧))
9		fvres 6901	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd ‘𝑧))
10	9	mpteq2ia 5204	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧))
11		vex 3457	. . . . 5 ⊢ 𝑥 ∈ V
12		vex 3457	. . . . 5 ⊢ 𝑦 ∈ V
13	11, 12	op2ndd 8001	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
14	13	mpompt 7531	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦)
15	8, 10, 14	3eqtri 2789	. 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 23842	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
19	16, 17, 18	syl2anc 596	. 2 ⊢ (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
20	15, 19	eqeltrrid 2867	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902   ↦ cmpt 5190   × cxp 5657   ↾ cres 5661   Fn wfn 6532  –onto→wfo 6535  ‘cfv 6537  (class class class)co 7417   ∈ cmpo 7419  2^nd c2nd 7989  TopOnctopon 23141   Cn ccn 23455   ×_t ctx 23792

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-map 8832  df-topgen 17534  df-top 23125  df-topon 23142  df-bases 23177  df-cn 23458  df-tx 23794

This theorem is used by:  cnmptcom  23910  xkofvcn  23916  cnmptk2  23918  txhmeo  24035  txswaphmeo  24037  ptunhmeo  24040  xkohmeo  24047  tgpsubcn  24322  istgp2  24323  oppgtmd  24329  prdstmdd  24356  dvrcn  24416  divcn  25102  cnrehmeo  25187  htpycom  25210  htpyco1  25212  htpycc  25214  reparphti  25231  pcohtpylem  25253  pcorevlem  25260  cxpcn  26990  vmcn  31188  dipcn  31209  mndpluscn  34444  cvxsconn  35830  cvmlift2lem6  35895