Theorem resima 5070

Description: A restriction to an image. (Contributed by NM, 29-Sep-2004.)

Assertion

Ref	Expression
resima	⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵)

Proof of Theorem resima

Step	Hyp	Ref	Expression
1		residm 5069	. . 3 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵)
2	1	rneqi 4984	. 2 ⊢ ran ((𝐴 ↾ 𝐵) ↾ 𝐵) = ran (𝐴 ↾ 𝐵)
3		df-ima 4761	. 2 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = ran ((𝐴 ↾ 𝐵) ↾ 𝐵)
4		df-ima 4761	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2263	1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵)

Syntax hints:   = wceq 1398  ran crn 4749   ↾ cres 4750   “ cima 4751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-xp 4754  df-rel 4755  df-cnv 4756  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761

This theorem is referenced by:  isarep2  5442  f1imacnv  5630  foimacnv  5631  djudm  7395  suplocexprlemell  8027  elq  9953  qnnen  13174