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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 6501 | . 2 ⊢ (Fun 𝐹 → Fun ◡◡𝐹) | |
2 | imadif 6518 | . 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 1539 ∖ cdif 3884 ◡ccnv 5588 “ cima 5592 Fun wfun 6427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-fun 6435 |
This theorem is referenced by: gsumpropd2lem 18363 supppreima 31025 elrspunidl 31606 fsumcvg4 31900 zrhunitpreima 31928 imambfm 32229 carsggect 32285 sibfof 32307 eulerpartlemmf 32342 itg2addnclem 35828 itg2addnclem2 35829 smfresal 44322 |
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