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Theorem feqresmptf 44508
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 2897 . . 3 𝑥𝐶
2 feqresmptf.1 . . . 4 𝑥𝐹
32, 1nfres 5977 . . 3 𝑥(𝐹𝐶)
4 feqresmptf.2 . . . 4 (𝜑𝐹:𝐴𝐵)
5 feqresmptf.3 . . . 4 (𝜑𝐶𝐴)
64, 5fssresd 6752 . . 3 (𝜑 → (𝐹𝐶):𝐶𝐵)
71, 3, 6feqmptdf 6956 . 2 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)))
8 fvres 6904 . . 3 (𝑥𝐶 → ((𝐹𝐶)‘𝑥) = (𝐹𝑥))
98mpteq2ia 5244 . 2 (𝑥𝐶 ↦ ((𝐹𝐶)‘𝑥)) = (𝑥𝐶 ↦ (𝐹𝑥))
107, 9eqtrdi 2782 1 (𝜑 → (𝐹𝐶) = (𝑥𝐶 ↦ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wnfc 2877  wss 3943  cmpt 5224  cres 5671  wf 6533  cfv 6537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545
This theorem is referenced by: (None)
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