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Theorem fnresin1 5902
Description: Restriction of a function's domain with an intersection. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
fnresin1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))

Proof of Theorem fnresin1
StepHypRef Expression
1 inss1 3791 . 2 (𝐴𝐵) ⊆ 𝐴
2 fnssres 5901 . 2 ((𝐹 Fn 𝐴 ∧ (𝐴𝐵) ⊆ 𝐴) → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
31, 2mpan2 702 1 (𝐹 Fn 𝐴 → (𝐹 ↾ (𝐴𝐵)) Fn (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  cin 3535  wss 3536  cres 5027   Fn wfn 5782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-br 4575  df-opab 4635  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-res 5037  df-fun 5789  df-fn 5790
This theorem is referenced by:  wfrlem4  7279  fnresin  28615  frrlem4  30830
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