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Theorem funimass1 4919
Description: A kind of contraposition law that infers a subclass of an image from a preimage subclass. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass1 ((Fun 𝐹 A ⊆ ran 𝐹) → ((𝐹A) ⊆ BA ⊆ (𝐹B)))

Proof of Theorem funimass1
StepHypRef Expression
1 imass2 4644 . 2 ((𝐹A) ⊆ B → (𝐹 “ (𝐹A)) ⊆ (𝐹B))
2 funimacnv 4918 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐹A)) = (A ∩ ran 𝐹))
3 dfss 2926 . . . . . 6 (A ⊆ ran 𝐹A = (A ∩ ran 𝐹))
43biimpi 113 . . . . 5 (A ⊆ ran 𝐹A = (A ∩ ran 𝐹))
54eqcomd 2042 . . . 4 (A ⊆ ran 𝐹 → (A ∩ ran 𝐹) = A)
62, 5sylan9eq 2089 . . 3 ((Fun 𝐹 A ⊆ ran 𝐹) → (𝐹 “ (𝐹A)) = A)
76sseq1d 2966 . 2 ((Fun 𝐹 A ⊆ ran 𝐹) → ((𝐹 “ (𝐹A)) ⊆ (𝐹B) ↔ A ⊆ (𝐹B)))
81, 7syl5ib 143 1 ((Fun 𝐹 A ⊆ ran 𝐹) → ((𝐹A) ⊆ BA ⊆ (𝐹B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  cin 2910  wss 2911  ccnv 4287  ran crn 4289  cima 4291  Fun wfun 4839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847
This theorem is referenced by: (None)
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