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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 6827 | . . 3 ⊢ (Fun 𝐹 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵))) | |
2 | df-ss 3945 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴) | |
3 | 2 | biimpi 218 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐵) = 𝐴) |
4 | 3 | imaeq2d 5922 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (◡𝐹 “ (𝐴 ∩ 𝐵)) = (◡𝐹 “ 𝐴)) |
5 | 1, 4 | sylan9req 2876 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴)) |
6 | df-ss 3945 | . 2 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ 𝐵)) = (◡𝐹 “ 𝐴)) | |
7 | 5, 6 | sylibr 236 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ 𝐵) → (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∩ cin 3928 ⊆ wss 3929 ◡ccnv 5547 “ cima 5551 Fun wfun 6342 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-fun 6350 |
This theorem is referenced by: pwrssmgc 30680 carsggect 31597 eulerpartlemmf 31654 eulerpartlemgf 31658 orvclteinc 31754 |
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