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Theorem funss 6521
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 5738 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
2 coss1 5812 . . . . 5 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐴))
3 cnvss 5829 . . . . . 6 (𝐴𝐵𝐴𝐵)
4 coss2 5813 . . . . . 6 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
53, 4syl 17 . . . . 5 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
62, 5sstrd 3955 . . . 4 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐵))
7 sstr2 3952 . . . 4 ((𝐴𝐴) ⊆ (𝐵𝐵) → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
86, 7syl 17 . . 3 (𝐴𝐵 → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
91, 8anim12d 610 . 2 (𝐴𝐵 → ((Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I )))
10 df-fun 6499 . 2 (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ))
11 df-fun 6499 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
129, 10, 113imtr4g 296 1 (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wss 3911   I cid 5531  ccnv 5633  ccom 5638  Rel wrel 5639  Fun wfun 6491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3446  df-in 3918  df-ss 3928  df-br 5107  df-opab 5169  df-rel 5641  df-cnv 5642  df-co 5643  df-fun 6499
This theorem is referenced by:  funeq  6522  funopab4  6539  funres  6544  fun0  6567  funcnvcnv  6569  funin  6578  funres11  6579  foimacnv  6802  funelss  7980  funsssuppss  8122  strle1  17035  strssd  17083  pjpm  21130  subgrfun  28271  setrecsss  47232
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