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Theorem funimass2 5930
Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. (Contributed by NM, 25-May-2004.)
Assertion
Ref Expression
funimass2 ((Fun 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)

Proof of Theorem funimass2
StepHypRef Expression
1 imass2 5460 . 2 (𝐴 ⊆ (𝐹𝐵) → (𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)))
2 funimacnv 5928 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐹𝐵)) = (𝐵 ∩ ran 𝐹))
32sseq2d 3612 . . . 4 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) ↔ (𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹)))
4 inss1 3811 . . . . 5 (𝐵 ∩ ran 𝐹) ⊆ 𝐵
5 sstr2 3590 . . . . 5 ((𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹𝐴) ⊆ 𝐵))
64, 5mpi 20 . . . 4 ((𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹𝐴) ⊆ 𝐵)
73, 6syl6bi 243 . . 3 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵))
87imp 445 . 2 ((Fun 𝐹 ∧ (𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵))) → (𝐹𝐴) ⊆ 𝐵)
91, 8sylan2 491 1 ((Fun 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  cin 3554  wss 3555  ccnv 5073  ran crn 5075  cima 5077  Fun wfun 5841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-fun 5849
This theorem is referenced by:  fvimacnvi  6287  lmhmlsp  18968  2ndcomap  21171  tgqtop  21425  kqreglem1  21454  fmfnfmlem4  21671  fmucnd  22006  cfilucfil  22274
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