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Theorem f1ss 5442
Description: A function that is one-to-one is also one-to-one on some superset of its range. (Contributed by Mario Carneiro, 12-Jan-2013.)
Assertion
Ref Expression
f1ss  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A -1-1-> C )

Proof of Theorem f1ss
StepHypRef Expression
1 f1f 5437 . . 3  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fss 5397 . . 3  |-  ( ( F : A --> B  /\  B  C_  C )  ->  F : A --> C )
31, 2sylan 457 . 2  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A --> C )
4 df-f1 5260 . . . 4  |-  ( F : A -1-1-> B  <->  ( F : A --> B  /\  Fun  `' F ) )
54simprbi 450 . . 3  |-  ( F : A -1-1-> B  ->  Fun  `' F )
65adantr 451 . 2  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  Fun  `' F
)
7 df-f1 5260 . 2  |-  ( F : A -1-1-> C  <->  ( F : A --> C  /\  Fun  `' F ) )
83, 6, 7sylanbrc 645 1  |-  ( ( F : A -1-1-> B  /\  B  C_  C )  ->  F : A -1-1-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    C_ wss 3152   `'ccnv 4688   Fun wfun 5249   -->wf 5251   -1-1->wf1 5252
This theorem is referenced by:  domssex2  7021  1sdom  7065  marypha1lem  7186  marypha2  7192  isinffi  7625  fseqenlem1  7651  dfac12r  7772  ackbij2  7869  cff1  7884  fin23lem28  7966  fin23lem41  7978  pwfseqlem5  8285  hashf1lem1  11393  gsumzres  15194  gsumzcl  15195  gsumzf1o  15196  gsumzaddlem  15203  gsumzmhm  15210  gsumzoppg  15216  dvne0f1  19359  erdsze2lem1  23734  eldioph2lem2  26840  eldioph2  26841  lindfres  27293  islindf3  27296  usisuslgra  28113  uslgra1  28118  usgra1  28119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-in 3159  df-ss 3166  df-f 5259  df-f1 5260
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