ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  funimass2 GIF version

Theorem funimass2 5306
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 5016 . 2 (𝐴 ⊆ (𝐹𝐵) → (𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)))
2 funimacnv 5304 . . . . 5 (Fun 𝐹 → (𝐹 “ (𝐹𝐵)) = (𝐵 ∩ ran 𝐹))
32sseq2d 3197 . . . 4 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) ↔ (𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹)))
4 inss1 3367 . . . . 5 (𝐵 ∩ ran 𝐹) ⊆ 𝐵
5 sstr2 3174 . . . . 5 ((𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹) → ((𝐵 ∩ ran 𝐹) ⊆ 𝐵 → (𝐹𝐴) ⊆ 𝐵))
64, 5mpi 15 . . . 4 ((𝐹𝐴) ⊆ (𝐵 ∩ ran 𝐹) → (𝐹𝐴) ⊆ 𝐵)
73, 6biimtrdi 163 . . 3 (Fun 𝐹 → ((𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵))
87imp 124 . 2 ((Fun 𝐹 ∧ (𝐹𝐴) ⊆ (𝐹 “ (𝐹𝐵))) → (𝐹𝐴) ⊆ 𝐵)
91, 8sylan2 286 1 ((Fun 𝐹𝐴 ⊆ (𝐹𝐵)) → (𝐹𝐴) ⊆ 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  cin 3140  wss 3141  ccnv 4637  ran crn 4639  cima 4641  Fun wfun 5222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-fun 5230
This theorem is referenced by:  fvimacnvi  5643
  Copyright terms: Public domain W3C validator