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Theorem dffn5f 6715
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 6703 . 2 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
2 dffn5f.1 . . . . 5 𝑥𝐹
3 nfcv 2958 . . . . 5 𝑥𝑧
42, 3nffv 6659 . . . 4 𝑥(𝐹𝑧)
5 nfcv 2958 . . . 4 𝑧(𝐹𝑥)
6 fveq2 6649 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
74, 5, 6cbvmpt 5134 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑥𝐴 ↦ (𝐹𝑥))
87eqeq2i 2814 . 2 (𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
91, 8bitri 278 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wnfc 2939  cmpt 5113   Fn wfn 6323  cfv 6328
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
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 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  prdsgsum  19098  lgamgulm2  25625  fcomptf  30425  esumsup  31462  poimirlem16  35072  poimirlem19  35075  refsum2cnlem1  41663  etransclem2  42875
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