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Theorem constmap 36160
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 (𝐵𝐶 → (𝐴 × {𝐵}) ∈ (𝐶𝑚 𝐴))

Proof of Theorem constmap
StepHypRef Expression
1 fconst6g 5890 . 2 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴𝐶)
2 constmap.3 . . 3 𝐶 ∈ V
3 constmap.1 . . 3 𝐴 ∈ V
42, 3elmap 7646 . 2 ((𝐴 × {𝐵}) ∈ (𝐶𝑚 𝐴) ↔ (𝐴 × {𝐵}):𝐴𝐶)
51, 4sylibr 222 1 (𝐵𝐶 → (𝐴 × {𝐵}) ∈ (𝐶𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1938  Vcvv 3077  {csn 4028   × cxp 4930  wf 5685  (class class class)co 6425  𝑚 cmap 7618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6721
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-map 7620
This theorem is referenced by:  mzpclall  36174  mzpindd  36193
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