Theorem cnmpt2c 23808

Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)

Hypotheses

Ref	Expression
cnmpt21.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
cnmpt21.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
cnmpt2c.l	⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
cnmpt2c.p	⊢ (𝜑 → 𝑃 ∈ 𝑍)

Assertion

Ref	Expression
cnmpt2c	⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Distinct variable groups:   𝑥,𝑦,𝐿   𝜑,𝑥,𝑦   𝑥,𝑋,𝑦   𝑥,𝑃,𝑦   𝑥,𝑌,𝑦   𝑥,𝑍,𝑦

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

Proof of Theorem cnmpt2c

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2764	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑃 = 𝑃)
2	1	mpompt 7526	. 2 ⊢ (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃)
3		cnmpt21.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
4		cnmpt21.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
5		txtopon 23729	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6	3, 4, 5	syl2anc 595	. . 3 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
7		cnmpt2c.l	. . 3 ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑍))
8		cnmpt2c.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝑍)
9	6, 7, 8	cnmptc 23800	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
10	2, 9	eqeltrrid 2868	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ⟨cop 4596   ↦ cmpt 5193   × cxp 5661  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  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-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-cnp 23366  df-tx 23700

This theorem is referenced by:  cnrehmeo  25093  pcopt  25162  pcopt2  25163  vmcn  31032  dipcn  31053  cvxsconn  35716