Theorem fnresdm 2731

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 2596	. 2 |- ((Rel F /\ dom F (_ A) -> (F |` A) = F)
2		fnrel 2722	. 2 |- (F Fn A -> Rel F)
3		fndm 2723	. . 3 |- (F Fn A -> dom F = A)
4		eqimss 1548	. . 3 |- (dom F = A -> dom F (_ A)
5	3, 4	syl 12	. 2 |- (F Fn A -> dom F (_ A)
6	1, 2, 5	sylanc 361	1 |- (F Fn A -> (F |` A) = F)

Syntax hints:   -> wi 2   = wceq 1091   (_ wss 1487  dom cdm 2410   |` cres 2412  Rel wrel 2415   Fn wfn 2417

This theorem is referenced by:  abianfp 3000  mapunen 3397

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-dm 2428  df-res 2430  df-fun 2432  df-fn 2433

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