Theorem cnmpt2nd 23807

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 8008	. . . . . 6 ⊢ 2nd :V–onto→V
2		fofn 6796	. . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 2nd Fn V
4		ssv 3962	. . . . 5 ⊢ (𝑋 × 𝑌) ⊆ V
5		fnssres 6660	. . . . 5 ⊢ ((2nd Fn V ∧ (𝑋 × 𝑌) ⊆ V) → (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌))
6	3, 4, 5	mp2an 704	. . . 4 ⊢ (2nd ↾ (𝑋 × 𝑌)) Fn (𝑋 × 𝑌)
7		dffn5 6941	. . . 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 6902	. . . 4 ⊢ (𝑧 ∈ (𝑋 × 𝑌) → ((2nd ↾ (𝑋 × 𝑌))‘𝑧) = (2nd ‘𝑧))
10	9	mpteq2ia 5207	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ ((2nd ↾ (𝑋 × 𝑌))‘𝑧)) = (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧))
11		vex 3459	. . . . 5 ⊢ 𝑥 ∈ V
12		vex 3459	. . . . 5 ⊢ 𝑦 ∈ V
13	11, 12	op2ndd 7998	. . . 4 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (2nd ‘𝑧) = 𝑦)
14	13	mpompt 7526	. . 3 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ (2nd ‘𝑧)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦)
15	8, 10, 14	3eqtri 2790	. 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 23748	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
19	16, 17, 18	syl2anc 595	. 2 ⊢ (𝜑 → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))
20	15, 19	eqeltrrid 2868	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑦) ∈ ((𝐽 ×t 𝐾) Cn 𝐾))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ⊆ wss 3906   ↦ cmpt 5193   × cxp 5661   ↾ cres 5665   Fn wfn 6533  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  2^nd c2nd 7986  TopOnctopon 23048   Cn ccn 23362   ×_t ctx 23698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-map 8827  df-topgen 17497  df-top 23032  df-topon 23049  df-bases 23084  df-cn 23365  df-tx 23700

This theorem is referenced by:  cnmptcom  23816  xkofvcn  23822  cnmptk2  23824  txhmeo  23941  txswaphmeo  23943  ptunhmeo  23946  xkohmeo  23953  tgpsubcn  24228  istgp2  24229  oppgtmd  24235  prdstmdd  24262  dvrcn  24322  divcn  25008  cnrehmeo  25093  htpycom  25116  htpyco1  25118  htpycc  25120  reparphti  25137  pcohtpylem  25159  pcorevlem  25166  cxpcn  26891  vmcn  31032  dipcn  31053  mndpluscn  34297  cvxsconn  35716  cvmlift2lem6  35781