Theorem fnresdm 5008

Description: A function does not change when restricted to its domain. (Contributed by NM, 5-Sep-2004.)

Assertion

Ref	Expression
fnresdm	⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)

Proof of Theorem fnresdm

Step	Hyp	Ref	Expression
1		fnrel 4997	. 2 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹)
2		fndm 4998	. . 3 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
3		eqimss 2997	. . 3 ⊢ (dom 𝐹 = 𝐴 → dom 𝐹 ⊆ 𝐴)
4	2, 3	syl 14	. 2 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 ⊆ 𝐴)
5		relssres 4648	. 2 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ 𝐴) → (𝐹 ↾ 𝐴) = 𝐹)
6	1, 4, 5	syl2anc 391	1 ⊢ (𝐹 Fn 𝐴 → (𝐹 ↾ 𝐴) = 𝐹)

Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917  dom cdm 4345   ↾ cres 4347  Rel wrel 4350   Fn wfn 4897

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355  df-res 4357  df-fun 4904  df-fn 4905

This theorem is referenced by:  fnima  5017  fresin  5068  resasplitss  5069  fsnunfv  5363  fsnunres  5364  fseq1p1m1  8956