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Theorem funss 4834
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 4342 . . 3 (AB → (Rel B → Rel A))
2 coss1 4406 . . . . 5 (AB → (AA) ⊆ (BA))
3 cnvss 4423 . . . . . 6 (ABAB)
4 coss2 4407 . . . . . 6 (AB → (BA) ⊆ (BB))
53, 4syl 14 . . . . 5 (AB → (BA) ⊆ (BB))
62, 5sstrd 2923 . . . 4 (AB → (AA) ⊆ (BB))
7 sstr2 2920 . . . 4 ((AA) ⊆ (BB) → ((BB) ⊆ I → (AA) ⊆ I ))
86, 7syl 14 . . 3 (AB → ((BB) ⊆ I → (AA) ⊆ I ))
91, 8anim12d 318 . 2 (AB → ((Rel B (BB) ⊆ I ) → (Rel A (AA) ⊆ I )))
10 df-fun 4819 . 2 (Fun B ↔ (Rel B (BB) ⊆ I ))
11 df-fun 4819 . 2 (Fun A ↔ (Rel A (AA) ⊆ I ))
129, 10, 113imtr4g 194 1 (AB → (Fun B → Fun A))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wss 2885   I cid 3988  ccnv 4259  ccom 4264  Rel wrel 4265  Fun wfun 4811
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995
This theorem depends on definitions:  df-bi 110  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-in 2892  df-ss 2899  df-br 3728  df-opab 3782  df-rel 4267  df-cnv 4268  df-co 4269  df-fun 4819
This theorem is referenced by:  funeq  4835  funopab4  4851  funres  4855  fun0  4871  funcnvcnv  4872  funin  4884  funres11  4885  foimacnv  5057  tfrlemibfn  5851
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