Theorem resindm 4869

Description: When restricting a relation, intersecting with the domain of the relation has no effect. (Contributed by FL, 6-Oct-2008.)

Assertion

Ref	Expression
resindm	⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))

Proof of Theorem resindm

Step	Hyp	Ref	Expression
1		resdm 4866	. . 3 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
2	1	ineq2d 3282	. 2 ⊢ (Rel 𝐴 → ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ 𝐴))
3		resindi 4842	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴))
4		incom 3273	. . 3 ⊢ ((𝐴 ↾ 𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴 ↾ 𝐵))
5		inres 4844	. . 3 ⊢ (𝐴 ∩ (𝐴 ↾ 𝐵)) = ((𝐴 ∩ 𝐴) ↾ 𝐵)
6		inidm 3290	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	6	reseq1i 4823	. . 3 ⊢ ((𝐴 ∩ 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵)
8	4, 5, 7	3eqtrri 2166	. 2 ⊢ (𝐴 ↾ 𝐵) = ((𝐴 ↾ 𝐵) ∩ 𝐴)
9	2, 3, 8	3eqtr4g 2198	1 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1332   ∩ cin 3075  dom cdm 4547   ↾ cres 4549  Rel wrel 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-dm 4557  df-res 4559

This theorem is referenced by:  resdmdfsn  4870