![]() |
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 6621 | . 2 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
2 | imadif 6638 | . 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 1533 ∖ cdif 3941 ◡ccnv 5677 “ cima 5681 Fun wfun 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-fun 6551 |
This theorem is referenced by: gsumpropd2lem 18642 supppreima 32553 elrspunidl 33240 fsumcvg4 33682 zrhunitpreima 33710 imambfm 34013 carsggect 34069 sibfof 34091 eulerpartlemmf 34126 itg2addnclem 37275 itg2addnclem2 37276 smfresal 46314 |
Copyright terms: Public domain | W3C validator |