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Theorem feqresmpt 5170
Description: Express a restricted function as a mapping. (Contributed by Mario Carneiro, 18-May-2016.)
Hypotheses
Ref Expression
feqmptd.1 (φ𝐹:AB)
feqresmpt.2 (φ𝐶A)
Assertion
Ref Expression
feqresmpt (φ → (𝐹𝐶) = (x 𝐶 ↦ (𝐹x)))
Distinct variable groups:   x,A   x,𝐶   x,𝐹
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem feqresmpt
StepHypRef Expression
1 feqmptd.1 . . . 4 (φ𝐹:AB)
2 feqresmpt.2 . . . 4 (φ𝐶A)
3 fssres 5009 . . . 4 ((𝐹:AB 𝐶A) → (𝐹𝐶):𝐶B)
41, 2, 3syl2anc 391 . . 3 (φ → (𝐹𝐶):𝐶B)
54feqmptd 5169 . 2 (φ → (𝐹𝐶) = (x 𝐶 ↦ ((𝐹𝐶)‘x)))
6 fvres 5141 . . 3 (x 𝐶 → ((𝐹𝐶)‘x) = (𝐹x))
76mpteq2ia 3834 . 2 (x 𝐶 ↦ ((𝐹𝐶)‘x)) = (x 𝐶 ↦ (𝐹x))
85, 7syl6eq 2085 1 (φ → (𝐹𝐶) = (x 𝐶 ↦ (𝐹x)))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  wss 2911  cmpt 3809  cres 4290  wf 4841  cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853
This theorem is referenced by: (None)
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