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Theorem funimass2 6584
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 funimacnv 6582 . . . 4 (Fun 𝐹 → (𝐹 “ (𝐹𝐵)) = (𝐵 ∩ ran 𝐹))
21sseq2d 3976 . . 3 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) ↔ (𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹)))
3 inss1 4188 . . . 4 (𝐵 ∩ ran 𝐹) ⊆ 𝐵
4 sstr2 3951 . . . 4 ((𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹𝐴) ⊆ 𝐵))
53, 4mpi 20 . . 3 ((𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹𝐴) ⊆ 𝐵)
62, 5syl6bi 252 . 2 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵))
7 imass2 6054 . 2 (𝐴 ⊆ (𝐹𝐵) → (𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)))
86, 7impel 506 1 ((Fun 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  cin 3909  wss 3910  ccnv 5632  ran crn 5634  cima 5636  Fun wfun 6490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-fun 6498
This theorem is referenced by:  fvimacnvi  7002  lmhmlsp  20508  2ndcomap  22807  tgqtop  23061  kqreglem1  23090  fmfnfmlem4  23306  fmucnd  23642  cfilucfil  23913  zarcmplem  32453
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