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Theorem fresin 5011
Description: An identity for the mapping relationship under restriction. (Contributed by Scott Fenton, 4-Sep-2011.) (Proof shortened by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
fresin (𝐹:AB → (𝐹𝑋):(A𝑋)⟶B)

Proof of Theorem fresin
StepHypRef Expression
1 inss1 3151 . . 3 (A𝑋) ⊆ A
2 fssres 5009 . . 3 ((𝐹:AB (A𝑋) ⊆ A) → (𝐹 ↾ (A𝑋)):(A𝑋)⟶B)
31, 2mpan2 401 . 2 (𝐹:AB → (𝐹 ↾ (A𝑋)):(A𝑋)⟶B)
4 resres 4567 . . . 4 ((𝐹A) ↾ 𝑋) = (𝐹 ↾ (A𝑋))
5 ffn 4989 . . . . . 6 (𝐹:AB𝐹 Fn A)
6 fnresdm 4951 . . . . . 6 (𝐹 Fn A → (𝐹A) = 𝐹)
75, 6syl 14 . . . . 5 (𝐹:AB → (𝐹A) = 𝐹)
87reseq1d 4554 . . . 4 (𝐹:AB → ((𝐹A) ↾ 𝑋) = (𝐹𝑋))
94, 8syl5eqr 2083 . . 3 (𝐹:AB → (𝐹 ↾ (A𝑋)) = (𝐹𝑋))
109feq1d 4977 . 2 (𝐹:AB → ((𝐹 ↾ (A𝑋)):(A𝑋)⟶B ↔ (𝐹𝑋):(A𝑋)⟶B))
113, 10mpbid 135 1 (𝐹:AB → (𝐹𝑋):(A𝑋)⟶B)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242  cin 2910  wss 2911  cres 4290   Fn wfn 4840  wf 4841
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-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by: (None)
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