Theorem resima 2595

Description: A restriction to an image.

Assertion

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

Proof of Theorem resima

Step	Hyp	Ref	Expression
1		residm 2594	. . 3 ⊢ ((A ↾ B) ↾ B) = (A ↾ B)
2	1	rneqi 2556	. 2 ⊢ ran ((A ↾ B) ↾ B) = ran (A ↾ B)
3		df-ima 2431	. 2 ⊢ ((A ↾ B) “ B) = ran ((A ↾ B) ↾ B)
4		df-ima 2431	. 2 ⊢ (A “ B) = ran (A ↾ B)
5	2, 3, 4	3eqtr4 1126	1 ⊢ ((A ↾ B) “ B) = (A “ B)

Syntax hints:   = wceq 1091  ran crn 2411   ↾ cres 2412   “ cima 2413

This theorem is referenced by:  f1imacnv 2814

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

metamath.org