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Theorem funss 6087
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 5376 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
2 coss1 5446 . . . . 5 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐴))
3 cnvss 5463 . . . . . 6 (𝐴𝐵𝐴𝐵)
4 coss2 5447 . . . . . 6 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
53, 4syl 17 . . . . 5 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
62, 5sstrd 3771 . . . 4 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐵))
7 sstr2 3768 . . . 4 ((𝐴𝐴) ⊆ (𝐵𝐵) → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
86, 7syl 17 . . 3 (𝐴𝐵 → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
91, 8anim12d 602 . 2 (𝐴𝐵 → ((Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I )))
10 df-fun 6070 . 2 (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ))
11 df-fun 6070 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
129, 10, 113imtr4g 287 1 (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wss 3732   I cid 5184  ccnv 5276  ccom 5281  Rel wrel 5282  Fun wfun 6062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-in 3739  df-ss 3746  df-br 4810  df-opab 4872  df-rel 5284  df-cnv 5285  df-co 5286  df-fun 6070
This theorem is referenced by:  funeq  6088  funopab4  6105  funres  6110  fun0  6132  funcnvcnv  6134  funin  6143  funres11  6144  foimacnv  6337  funsssuppss  7524  strssd  16183  strle1  16247  xpsc0  16488  xpsc1  16489  pjpm  20328  subgrfun  26452  frrlem5c  32162  setrecsss  43048
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