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Theorem constmap 42701
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 6798 . 2 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴𝐶)
2 constmap.3 . . 3 𝐶 ∈ V
3 constmap.1 . . 3 𝐴 ∈ V
42, 3elmap 8910 . 2 ((𝐴 × {𝐵}) ∈ (𝐶m 𝐴) ↔ (𝐴 × {𝐵}):𝐴𝐶)
51, 4sylibr 234 1 (𝐵𝐶 → (𝐴 × {𝐵}) ∈ (𝐶m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  Vcvv 3478  {csn 4631   × cxp 5687  wf 6559  (class class class)co 7431  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867
This theorem is referenced by:  mzpclall  42715  mzpindd  42734
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