Theorem resindm 4926

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 4923	. . 3 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴)
2	1	ineq2d 3323	. 2 ⊢ (Rel 𝐴 → ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ 𝐴))
3		resindi 4899	. 2 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ((𝐴 ↾ 𝐵) ∩ (𝐴 ↾ dom 𝐴))
4		incom 3314	. . 3 ⊢ ((𝐴 ↾ 𝐵) ∩ 𝐴) = (𝐴 ∩ (𝐴 ↾ 𝐵))
5		inres 4901	. . 3 ⊢ (𝐴 ∩ (𝐴 ↾ 𝐵)) = ((𝐴 ∩ 𝐴) ↾ 𝐵)
6		inidm 3331	. . . 4 ⊢ (𝐴 ∩ 𝐴) = 𝐴
7	6	reseq1i 4880	. . 3 ⊢ ((𝐴 ∩ 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵)
8	4, 5, 7	3eqtrri 2191	. 2 ⊢ (𝐴 ↾ 𝐵) = ((𝐴 ↾ 𝐵) ∩ 𝐴)
9	2, 3, 8	3eqtr4g 2224	1 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵))

Syntax hints:   → wi 4   = wceq 1343   ∩ cin 3115  dom cdm 4604   ↾ cres 4606  Rel wrel 4609

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-dm 4614  df-res 4616

This theorem is referenced by:  resdmdfsn  4927