Theorem fnresdm 3596

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 3392	. 2 |- ((Rel F /\ dom F (_ A) -> (F |` A) = F)
2		fnrel 3586	. 2 |- (F Fn A -> Rel F)
3		fndm 3587	. . 3 |- (F Fn A -> dom F = A)
4		eqimss 2109	. . 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 956   (_ wss 2047  dom cdm 3170   |` cres 3172  Rel wrel 3175   Fn wfn 3177

This theorem is referenced by:  abianfp 3962  mapunen 4502  facnnt 6933  fac0 6934  subgres 8117  sspg 8387  ssps 8389  sspn 8395  hhsssh 9139  cayleylem3 10411

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-dm 3188  df-res 3190  df-fun 3192  df-fn 3193

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