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Theorem feqresmptf 41727
 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 2982 . . 3 𝑥𝐶
2 feqresmptf.1 . . . 4 𝑥𝐹
32, 1nfres 5842 . . 3 𝑥(𝐹𝐶)
4 feqresmptf.2 . . . 4 (𝜑𝐹:𝐴𝐵)
5 feqresmptf.3 . . . 4 (𝜑𝐶𝐴)
64, 5fssresd 6533 . . 3 (𝜑 → (𝐹𝐶):𝐶𝐵)
71, 3, 6feqmptdf 6723 . 2 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)))
8 fvres 6677 . . 3 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
98mpteq2ia 5143 . 2 (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)) = (𝑥𝐶 ↦ (𝐹𝑥))
107, 9syl6eq 2875 1 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538  Ⅎwnfc 2962   ⊆ wss 3919   ↦ cmpt 5132   ↾ cres 5544  ⟶wf 6339  ‘cfv 6343 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351 This theorem is referenced by: (None)
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