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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 2905 . . . . 5 𝑥𝑧
42, 3nffv 6916 . . . 4 𝑥(𝐹𝑧)
5 nfcv 2905 . . . 4 𝑧(𝐹𝑥)
6 fveq2 6906 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
74, 5, 6cbvmpt 5253 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑥𝐴 ↦ (𝐹𝑥))
87eqeq2i 2750 . 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 1540  wnfc 2890  cmpt 5225   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  prdsgsum  19999  lgamgulm2  27079  fcomptf  32668  esumsup  34090  poimirlem16  37643  poimirlem19  37646  pwsgprod  42554  refsum2cnlem1  45042  etransclem2  46251
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