Theorem constmap 43145

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 6729	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶)
2		constmap.3	. . 3 ⊢ 𝐶 ∈ V
3		constmap.1	. . 3 ⊢ 𝐴 ∈ V
4	2, 3	elmap 8819	. 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 2114  Vcvv 3429  {csn 4567   × cxp 5629  ⟶wf 6494  (class class class)co 7367   ↑_m cmap 8773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-map 8775

This theorem is referenced by:  mzpclall  43159  mzpindd  43178