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Theorem funimass1 6650
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 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))

Proof of Theorem funimass1
StepHypRef Expression
1 imass2 6123 . 2 ((𝐹𝐴) ⊆ 𝐵 → (𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵))
2 funimacnv 6649 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
3 dfss 3982 . . . . . 6 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
43biimpi 216 . . . . 5 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
54eqcomd 2741 . . . 4 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
62, 5sylan9eq 2795 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → (𝐹 “ (𝐹𝐴)) = 𝐴)
76sseq1d 4027 . 2 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵) ↔ 𝐴 ⊆ (𝐹𝐵)))
81, 7imbitrid 244 1 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  cin 3962  wss 3963  ccnv 5688  ran crn 5690  cima 5692  Fun wfun 6557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565
This theorem is referenced by:  kqnrmlem1  23767  hmeontr  23793  nrmhmph  23818  cnheiborlem  25000
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