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Theorem funimass1 6433
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 5963 . 2 ((𝐹𝐴) ⊆ 𝐵 → (𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵))
2 funimacnv 6432 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
3 dfss 3957 . . . . . 6 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
43biimpi 217 . . . . 5 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
54eqcomd 2832 . . . 4 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
62, 5sylan9eq 2881 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → (𝐹 “ (𝐹𝐴)) = 𝐴)
76sseq1d 4002 . 2 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵) ↔ 𝐴 ⊆ (𝐹𝐵)))
81, 7syl5ib 245 1 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  cin 3939  wss 3940  ccnv 5553  ran crn 5555  cima 5557  Fun wfun 6346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-fun 6354
This theorem is referenced by:  kqnrmlem1  22267  hmeontr  22293  nrmhmph  22318  cnheiborlem  23473
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