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Theorem funss 4948
 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 4455 . . 3 (𝐴𝐵 → (Rel 𝐵 → Rel 𝐴))
2 coss1 4519 . . . . 5 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐴))
3 cnvss 4536 . . . . . 6 (𝐴𝐵𝐴𝐵)
4 coss2 4520 . . . . . 6 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
53, 4syl 14 . . . . 5 (𝐴𝐵 → (𝐵𝐴) ⊆ (𝐵𝐵))
62, 5sstrd 2983 . . . 4 (𝐴𝐵 → (𝐴𝐴) ⊆ (𝐵𝐵))
7 sstr2 2980 . . . 4 ((𝐴𝐴) ⊆ (𝐵𝐵) → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
86, 7syl 14 . . 3 (𝐴𝐵 → ((𝐵𝐵) ⊆ I → (𝐴𝐴) ⊆ I ))
91, 8anim12d 322 . 2 (𝐴𝐵 → ((Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ) → (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I )))
10 df-fun 4932 . 2 (Fun 𝐵 ↔ (Rel 𝐵 ∧ (𝐵𝐵) ⊆ I ))
11 df-fun 4932 . 2 (Fun 𝐴 ↔ (Rel 𝐴 ∧ (𝐴𝐴) ⊆ I ))
129, 10, 113imtr4g 198 1 (𝐴𝐵 → (Fun 𝐵 → Fun 𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   ⊆ wss 2945   I cid 4053  ◡ccnv 4372   ∘ ccom 4377  Rel wrel 4378  Fun wfun 4924 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-in 2952  df-ss 2959  df-br 3793  df-opab 3847  df-rel 4380  df-cnv 4381  df-co 4382  df-fun 4932 This theorem is referenced by:  funeq  4949  funopab4  4965  funres  4969  fun0  4985  funcnvcnv  4986  funin  4998  funres11  4999  foimacnv  5172  tfrlemibfn  5973
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