Theorem constmap 42724

Description: A constant (represented without dummy variables) is an element of a function set.

Note: In the following development, we will be quite often quantifying over functions and points in N-dimensional space (which are equivalent to functions from an "index set"). Many of the following theorems exist to transfer standard facts about functions to elements of function sets. (Contributed by Stefan O'Rear, 30-Aug-2014.) (Revised by Stefan O'Rear, 5-May-2015.)

Hypotheses

Ref	Expression
constmap.1	⊢ 𝐴 ∈ V
constmap.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
constmap	⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) ∈ (𝐶 ↑m 𝐴))

Proof of Theorem constmap

Step	Hyp	Ref	Expression
1		fconst6g 6797	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶)
2		constmap.3	. . 3 ⊢ 𝐶 ∈ V
3		constmap.1	. . 3 ⊢ 𝐴 ∈ V
4	2, 3	elmap 8911	. 2 ⊢ ((𝐴 × {𝐵}) ∈ (𝐶 ↑m 𝐴) ↔ (𝐴 × {𝐵}):𝐴⟶𝐶)
5	1, 4	sylibr 234	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) ∈ (𝐶 ↑m 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2108  Vcvv 3480  {csn 4626   × cxp 5683  ⟶wf 6557  (class class class)co 7431   ↑_m cmap 8866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868

This theorem is referenced by:  mzpclall  42738  mzpindd  42757