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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sspreima | Structured version Visualization version GIF version |
Description: The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.) |
Ref | Expression |
---|---|
sspreima | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inpreima 6811 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) | |
2 | df-ss 3898 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
3 | 2 | biimpi 219 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) |
4 | 3 | imaeq2d 5896 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ 𝐴)) |
5 | 1, 4 | sylan9req 2854 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴)) |
6 | df-ss 3898 | . 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴)) | |
7 | 5, 6 | sylibr 237 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∩ cin 3880 ⊆ wss 3881 ◡ccnv 5518 “ cima 5522 Fun wfun 6318 |
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 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
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-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-fun 6326 |
This theorem is referenced by: pwrssmgc 30706 gsumhashmul 30741 elrspunidl 31014 carsggect 31686 eulerpartlemmf 31743 eulerpartlemgf 31747 orvclteinc 31843 |
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