| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > difpreima | Structured version Visualization version GIF version | ||
| Description: Preimage of a difference. (Contributed by Mario Carneiro, 14-Jun-2016.) |
| Ref | Expression |
|---|---|
| difpreima | ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∖ 𝐵)) = ((◡𝐹 “ 𝐴) ∖ (◡𝐹 “ 𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | funcnvcnv 6633 | . 2 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
| 2 | imadif 6650 | . 2 ⊢ (Fun ◡◡𝐹 → (◡𝐹 “ (𝐴 ∖ 𝐵)) = ((◡𝐹 “ 𝐴) ∖ (◡𝐹 “ 𝐵))) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∖ 𝐵)) = ((◡𝐹 “ 𝐴) ∖ (◡𝐹 “ 𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∖ cdif 3948 ◡ccnv 5684 “ cima 5688 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 |
| This theorem is referenced by: gsumpropd2lem 18692 supppreima 32700 elrgspnsubrunlem2 33252 elrspunidl 33456 fsumcvg4 33949 zrhunitpreima 33977 imambfm 34264 carsggect 34320 sibfof 34342 eulerpartlemmf 34377 itg2addnclem 37678 itg2addnclem2 37679 smfresal 46803 |
| Copyright terms: Public domain | W3C validator |