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Theorem dffn5imf 5372
 Description: Representation of a function in terms of its values. (Contributed by Jim Kingdon, 31-Dec-2018.)
Hypothesis
Ref Expression
dffn5imf.1 𝑥𝐹
Assertion
Ref Expression
dffn5imf (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem dffn5imf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 dffn5im 5363 . 2 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
2 dffn5imf.1 . . . 4 𝑥𝐹
3 nfcv 2229 . . . 4 𝑥𝑧
42, 3nffv 5328 . . 3 𝑥(𝐹𝑧)
5 nfcv 2229 . . 3 𝑧(𝐹𝑥)
6 fveq2 5318 . . 3 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
74, 5, 6cbvmpt 3939 . 2 (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑥𝐴 ↦ (𝐹𝑥))
81, 7syl6eq 2137 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1290  Ⅎwnfc 2216   ↦ cmpt 3905   Fn wfn 5023  ‘cfv 5028 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045 This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-fv 5036 This theorem is referenced by: (None)
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