Theorem constmap 39330

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 6568	. 2 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}):𝐴⟶𝐶)
2		constmap.3	. . 3 ⊢ 𝐶 ∈ V
3		constmap.1	. . 3 ⊢ 𝐴 ∈ V
4	2, 3	elmap 8435	. 2 ⊢ ((𝐴 × {𝐵}) ∈ (𝐶 ↑m 𝐴) ↔ (𝐴 × {𝐵}):𝐴⟶𝐶)
5	1, 4	sylibr 236	1 ⊢ (𝐵 ∈ 𝐶 → (𝐴 × {𝐵}) ∈ (𝐶 ↑m 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2114  Vcvv 3494  {csn 4567   × cxp 5553  ⟶wf 6351  (class class class)co 7156   ↑_m cmap 8406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408

This theorem is referenced by:  mzpclall  39344  mzpindd  39363