Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  funss Structured version   Visualization version   GIF version

Theorem funss 6142
 Description: Subclass theorem for function predicate. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
Assertion
Ref Expression
funss (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))

Proof of Theorem funss
StepHypRef Expression
1 relss 5441 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
2 coss1 5510 . . . . 5 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐴))
3 cnvss 5527 . . . . . 6 (𝐴𝐵𝐴𝐵)
4 coss2 5511 . . . . . 6 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
53, 4syl 17 . . . . 5 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
62, 5sstrd 3837 . . . 4 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐵))
7 sstr2 3834 . . . 4 ((𝐴𝐴) ⊆ (𝐵𝐵) → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
86, 7syl 17 . . 3 (𝐴𝐵 → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
91, 8anim12d 604 . 2 (𝐴𝐵 → ((Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I )))
10 df-fun 6125 . 2 (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ))
11 df-fun 6125 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
129, 10, 113imtr4g 288 1 (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ⊆ wss 3798   I cid 5249  ◡ccnv 5341   ∘ ccom 5346  Rel wrel 5347  Fun wfun 6117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-in 3805  df-ss 3812  df-br 4874  df-opab 4936  df-rel 5349  df-cnv 5350  df-co 5351  df-fun 6125 This theorem is referenced by:  funeq  6143  funopab4  6160  funres  6165  fun0  6187  funcnvcnv  6189  funin  6198  funres11  6199  foimacnv  6395  funsssuppss  7586  strssd  16272  strle1  16332  xpsc0  16573  xpsc1  16574  pjpm  20415  subgrfun  26578  frrlem5c  32325  setrecsss  43342
 Copyright terms: Public domain W3C validator