Theorem cnmpt2c 23864

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 2767	. . 3 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝑃 = 𝑃)
2	1	mpompt 7537	. 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 23785	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6	3, 4, 5	syl2anc 596	. . 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 23856	. 2 ⊢ (𝜑 → (𝑧 ∈ (𝑋 × 𝑌) ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))
10	2, 9	eqeltrrid 2871	1 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝑃) ∈ ((𝐽 ×t 𝐾) Cn 𝐿))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ⟨cop 4600   ↦ cmpt 5197   × cxp 5664  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  TopOnctopon 23104   Cn ccn 23418   ×_t ctx 23754

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-1st 7995  df-2nd 7996  df-map 8835  df-topgen 17521  df-top 23088  df-topon 23105  df-bases 23140  df-cn 23421  df-cnp 23422  df-tx 23756

This theorem is used by:  cnrehmeo  25149  pcopt  25218  pcopt2  25219  vmcn  31088  dipcn  31109  cvxsconn  35756