Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > resima | Structured version Visualization version GIF version |
Description: A restriction to an image. (Contributed by NM, 29-Sep-2004.) |
Ref | Expression |
---|---|
resima | ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | residm 5886 | . . 3 ⊢ ((𝐴 ↾ 𝐵) ↾ 𝐵) = (𝐴 ↾ 𝐵) | |
2 | 1 | rneqi 5807 | . 2 ⊢ ran ((𝐴 ↾ 𝐵) ↾ 𝐵) = ran (𝐴 ↾ 𝐵) |
3 | df-ima 5568 | . 2 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = ran ((𝐴 ↾ 𝐵) ↾ 𝐵) | |
4 | df-ima 5568 | . 2 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
5 | 2, 3, 4 | 3eqtr4i 2854 | 1 ⊢ ((𝐴 ↾ 𝐵) “ 𝐵) = (𝐴 “ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ran crn 5556 ↾ cres 5557 “ cima 5558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 |
This theorem is referenced by: isarep2 6443 f1imacnv 6631 foimacnv 6632 dffv2 6756 islindf4 20982 qtopres 22306 |
Copyright terms: Public domain | W3C validator |