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Theorem dffn5imf 5549
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 5540 . 2 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
2 dffn5imf.1 . . . 4 𝑥𝐹
3 nfcv 2312 . . . 4 𝑥𝑧
42, 3nffv 5504 . . 3 𝑥(𝐹𝑧)
5 nfcv 2312 . . 3 𝑧(𝐹𝑥)
6 fveq2 5494 . . 3 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
74, 5, 6cbvmpt 4082 . 2 (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑥𝐴 ↦ (𝐹𝑥))
81, 7eqtrdi 2219 1 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wnfc 2299  cmpt 4048   Fn wfn 5191  cfv 5196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fn 5199  df-fv 5204
This theorem is referenced by: (None)
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