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Theorem dffn5f 6214
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 6203 . 2 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
2 dffn5f.1 . . . . 5 𝑥𝐹
3 nfcv 2761 . . . . 5 𝑥𝑧
42, 3nffv 6160 . . . 4 𝑥(𝐹𝑧)
5 nfcv 2761 . . . 4 𝑧(𝐹𝑥)
6 fveq2 6153 . . . 4 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
74, 5, 6cbvmpt 4714 . . 3 (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑥𝐴 ↦ (𝐹𝑥))
87eqeq2i 2633 . 2 (𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)) ↔ 𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
91, 8bitri 264 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wnfc 2748  cmpt 4678   Fn wfn 5847  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860
This theorem is referenced by:  prdsgsum  18309  lgamgulm2  24679  fcomptf  29323  esumsup  29956  poimirlem16  33092  poimirlem19  33095  refsum2cnlem1  38714  etransclem2  39786
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