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Mirrors > Home > MPE Home > Th. List > Mathboxes > imadifss | Structured version Visualization version GIF version |
Description: The difference of images is a subset of the image of the difference. (Contributed by Brendan Leahy, 21-Aug-2020.) |
Ref | Expression |
---|---|
imadifss | ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 4113 | . . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) | |
2 | undif2 4416 | . . . . 5 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴) | |
3 | 1, 2 | sseqtrri 3963 | . . . 4 ⊢ 𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵)) |
4 | imass2 6020 | . . . 4 ⊢ (𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵)) → (𝐹 “ 𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵)))) | |
5 | 3, 4 | ax-mp 5 | . . 3 ⊢ (𝐹 “ 𝐴) ⊆ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵))) |
6 | imaundi 6068 | . . 3 ⊢ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵))) | |
7 | 5, 6 | sseqtri 3962 | . 2 ⊢ (𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵))) |
8 | ssundif 4424 | . 2 ⊢ ((𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵))) ↔ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))) | |
9 | 7, 8 | mpbi 229 | 1 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3889 ∪ cun 3890 ⊆ wss 3892 “ cima 5603 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-12 2169 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-br 5082 df-opab 5144 df-xp 5606 df-cnv 5608 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 |
This theorem is referenced by: poimirlem30 35851 |
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