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Theorem dffn5f 6730
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 6718 . 2 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
2 dffn5f.1 . . . . 5 𝑥𝐹
3 nfcv 2977 . . . . 5 𝑥𝑧
42, 3nffv 6674 . . . 4 𝑥(𝐹𝑧)
5 nfcv 2977 . . . 4 𝑧(𝐹𝑥)
6 fveq2 6664 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
74, 5, 6cbvmpt 5159 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑥𝐴 ↦ (𝐹𝑥))
87eqeq2i 2834 . 2 (𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
91, 8bitri 276 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  wnfc 2961  cmpt 5138   Fn wfn 6344  cfv 6349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357
This theorem is referenced by:  prdsgsum  19032  lgamgulm2  25541  fcomptf  30332  esumsup  31248  poimirlem16  34790  poimirlem19  34793  refsum2cnlem1  41174  etransclem2  42402
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