Theorem resima 5044

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 5043	. . 3 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵)
2	1	rneqi 4958	. 2 ⊢ ran ((𝐴 ↾ 𝐵) ↾ 𝐵) = ran (𝐴 ↾ 𝐵)
3		df-ima 4736	. 2 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = ran ((𝐴 ↾ 𝐵) ↾ 𝐵)
4		df-ima 4736	. 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵)
5	2, 3, 4	3eqtr4i 2260	1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵)

Syntax hints:   = wceq 1395  ran crn 4724   ↾ cres 4725   “ cima 4726

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-rel 4730  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736

This theorem is referenced by:  isarep2  5414  f1imacnv  5597  foimacnv  5598  djudm  7298  suplocexprlemell  7926  elq  9849  qnnen  13045