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Theorem feqresmptf 42772
Description: Express a restricted function as a mapping. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
feqresmptf.1 𝑥𝐹
feqresmptf.2 (𝜑𝐹:𝐴𝐵)
feqresmptf.3 (𝜑𝐶𝐴)
Assertion
Ref Expression
feqresmptf (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
Distinct variable group:   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem feqresmptf
StepHypRef Expression
1 nfcv 2907 . . 3 𝑥𝐶
2 feqresmptf.1 . . . 4 𝑥𝐹
32, 1nfres 5893 . . 3 𝑥(𝐹𝐶)
4 feqresmptf.2 . . . 4 (𝜑𝐹:𝐴𝐵)
5 feqresmptf.3 . . . 4 (𝜑𝐶𝐴)
64, 5fssresd 6641 . . 3 (𝜑 → (𝐹𝐶):𝐶𝐵)
71, 3, 6feqmptdf 6839 . 2 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)))
8 fvres 6793 . . 3 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
98mpteq2ia 5177 . 2 (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)) = (𝑥𝐶 ↦ (𝐹𝑥))
107, 9eqtrdi 2794 1 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wnfc 2887  wss 3887  cmpt 5157  cres 5591  wf 6429  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441
This theorem is referenced by: (None)
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