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Mirrors > Home > MPE Home > Th. List > funimass2 | Structured version Visualization version GIF version |
Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.) |
Ref | Expression |
---|---|
funimass2 | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimacnv 6631 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 “ (◡𝐹 “ 𝐵)) = (𝐵 ∩ ran 𝐹)) | |
2 | 1 | sseq2d 4013 | . . 3 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) ↔ (𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹))) |
3 | inss1 4229 | . . . 4 ⊢ (𝐵 ∩ ran 𝐹) ⊆ 𝐵 | |
4 | sstr2 3987 | . . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹 “ 𝐴) ⊆ 𝐵)) | |
5 | 3, 4 | mpi 20 | . . 3 ⊢ ((𝐹 “ 𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹 “ 𝐴) ⊆ 𝐵) |
6 | 2, 5 | biimtrdi 252 | . 2 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵)) |
7 | imass2 6104 | . 2 ⊢ (𝐴 ⊆ (◡𝐹 “ 𝐵) → (𝐹 “ 𝐴) ⊆ (𝐹 “ (◡𝐹 “ 𝐵))) | |
8 | 6, 7 | impel 504 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ (◡𝐹 “ 𝐵)) → (𝐹 “ 𝐴) ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∩ cin 3947 ⊆ wss 3948 ◡ccnv 5673 ran crn 5675 “ cima 5677 Fun wfun 6539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3466 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4325 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-br 5146 df-opab 5208 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-fun 6547 |
This theorem is referenced by: fvimacnvi 7056 lmhmlsp 21022 2ndcomap 23449 tgqtop 23703 kqreglem1 23732 fmfnfmlem4 23948 fmucnd 24284 cfilucfil 24555 zarcmplem 33708 |
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