Theorem fnresdm 3588

Description: A function does not change when restricted to its domain.

Assertion

Ref	Expression
fnresdm	|- (F Fn A -> (F |` A) = F)

Proof of Theorem fnresdm

Step	Hyp	Ref	Expression
1		relssres 3384	. 2 |- ((Rel F /\ dom F (_ A) -> (F |` A) = F)
2		fnrel 3578	. 2 |- (F Fn A -> Rel F)
3		fndm 3579	. . 3 |- (F Fn A -> dom F = A)
4		eqimss 2105	. . 3 |- (dom F = A -> dom F (_ A)
5	3, 4	syl 10	. 2 |- (F Fn A -> dom F (_ A)
6	1, 2, 5	sylanc 471	1 |- (F Fn A -> (F |` A) = F)

Syntax hints:   -> wi 3   = wceq 954   (_ wss 2043  dom cdm 3165   |` cres 3167  Rel wrel 3170   Fn wfn 3172

This theorem is referenced by:  abianfp 3953  mapunen 4488  facnnt 6878  fac0 6879  subgres 8069  sspg 8334  ssps 8336  sspn 8342  hhsssh 9078  cayleylem3 10345

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-xp 3179  df-rel 3180  df-dm 3183  df-res 3185  df-fun 3187  df-fn 3188

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