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Mathbox for Brendan Leahy |
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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 4100 | . . . . 5 ⊢ 𝐴 ⊆ (𝐵 ∪ 𝐴) | |
2 | undif2 4383 | . . . . 5 ⊢ (𝐵 ∪ (𝐴 ∖ 𝐵)) = (𝐵 ∪ 𝐴) | |
3 | 1, 2 | sseqtrri 3952 | . . . 4 ⊢ 𝐴 ⊆ (𝐵 ∪ (𝐴 ∖ 𝐵)) |
4 | imass2 5932 | . . . 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 5975 | . . 3 ⊢ (𝐹 “ (𝐵 ∪ (𝐴 ∖ 𝐵))) = ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵))) | |
7 | 5, 6 | sseqtri 3951 | . 2 ⊢ (𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵))) |
8 | ssundif 4391 | . 2 ⊢ ((𝐹 “ 𝐴) ⊆ ((𝐹 “ 𝐵) ∪ (𝐹 “ (𝐴 ∖ 𝐵))) ↔ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵))) | |
9 | 7, 8 | mpbi 233 | 1 ⊢ ((𝐹 “ 𝐴) ∖ (𝐹 “ 𝐵)) ⊆ (𝐹 “ (𝐴 ∖ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ∖ cdif 3878 ∪ cun 3879 ⊆ wss 3881 “ cima 5522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 |
This theorem is referenced by: poimirlem30 35087 |
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