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Theorem dffn5f 6980
Description: Representation of a function in terms of its values. (Contributed by Mario Carneiro, 3-Jul-2015.)
Hypothesis
Ref Expression
dffn5f.1 𝑥𝐹
Assertion
Ref Expression
dffn5f (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem dffn5f
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffn5 6967 . 2 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
2 dffn5f.1 . . . . 5 𝑥𝐹
3 nfcv 2903 . . . . 5 𝑥𝑧
42, 3nffv 6917 . . . 4 𝑥(𝐹𝑧)
5 nfcv 2903 . . . 4 𝑧(𝐹𝑥)
6 fveq2 6907 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
74, 5, 6cbvmpt 5259 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑥𝐴 ↦ (𝐹𝑥))
87eqeq2i 2748 . 2 (𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
91, 8bitri 275 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1537  wnfc 2888  cmpt 5231   Fn wfn 6558  cfv 6563
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-pr 5438
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-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  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-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  prdsgsum  20014  lgamgulm2  27094  fcomptf  32675  esumsup  34070  poimirlem16  37623  poimirlem19  37626  pwsgprod  42531  refsum2cnlem1  44975  etransclem2  46192
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