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Theorem funss 4863
 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 4370 . . 3 (AB → (Rel B → Rel A))
2 coss1 4434 . . . . 5 (AB → (AA) ⊆ (BA))
3 cnvss 4451 . . . . . 6 (ABAB)
4 coss2 4435 . . . . . 6 (AB → (BA) ⊆ (BB))
53, 4syl 14 . . . . 5 (AB → (BA) ⊆ (BB))
62, 5sstrd 2949 . . . 4 (AB → (AA) ⊆ (BB))
7 sstr2 2946 . . . 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 4847 . 2 (Fun B ↔ (Rel B (BB) ⊆ I ))
11 df-fun 4847 . 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 2911   I cid 4016  ◡ccnv 4287   ∘ ccom 4292  Rel wrel 4293  Fun wfun 4839 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-fun 4847 This theorem is referenced by:  funeq  4864  funopab4  4880  funres  4884  fun0  4900  funcnvcnv  4901  funin  4913  funres11  4914  foimacnv  5087  tfrlemibfn  5883
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