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Theorem constmap 39570
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 6558 . 2 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴𝐶)
2 constmap.3 . . 3 𝐶 ∈ V
3 constmap.1 . . 3 𝐴 ∈ V
42, 3elmap 8431 . 2 ((𝐴 × {𝐵}) ∈ (𝐶m 𝐴) ↔ (𝐴 × {𝐵}):𝐴𝐶)
51, 4sylibr 237 1 (𝐵𝐶 → (𝐴 × {𝐵}) ∈ (𝐶m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115  Vcvv 3480  {csn 4550   × cxp 5540  wf 6339  (class class class)co 7149  m cmap 8402
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404
This theorem is referenced by:  mzpclall  39584  mzpindd  39603
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