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Theorem funimass1 6584
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 6055 . 2 ((𝐹𝐴) ⊆ 𝐵 → (𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵))
2 funimacnv 6583 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐹𝐴)) = (𝐴 ∩ ran 𝐹))
3 dfss 3929 . . . . . 6 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
43biimpi 215 . . . . 5 (𝐴 ⊆ ran 𝐹𝐴 = (𝐴 ∩ ran 𝐹))
54eqcomd 2739 . . . 4 (𝐴 ⊆ ran 𝐹 → (𝐴 ∩ ran 𝐹) = 𝐴)
62, 5sylan9eq 2793 . . 3 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → (𝐹 “ (𝐹𝐴)) = 𝐴)
76sseq1d 3976 . 2 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹 “ (𝐹𝐴)) ⊆ (𝐹𝐵) ↔ 𝐴 ⊆ (𝐹𝐵)))
81, 7imbitrid 243 1 ((Fun 𝐹𝐴 ⊆ ran 𝐹) → ((𝐹𝐴) ⊆ 𝐵𝐴 ⊆ (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  cin 3910  wss 3911  ccnv 5633  ran crn 5635  cima 5637  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499
This theorem is referenced by:  kqnrmlem1  23110  hmeontr  23136  nrmhmph  23161  cnheiborlem  24333
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