MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1dom2g Unicode version

Theorem f1dom2g 6967
Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6969 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
f1dom2g  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  A  ~<_  B )

Proof of Theorem f1dom2g
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 f1f 5520 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 5484 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1215 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
433coml 1158 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F  e.  _V )
5 simp3 957 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
6 f1eq1 5515 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-> B  <->  F : A -1-1-> B ) )
76spcegv 2945 . . 3  |-  ( F  e.  _V  ->  ( F : A -1-1-> B  ->  E. f  f : A -1-1-> B ) )
84, 5, 7sylc 56 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  E. f  f : A -1-1-> B )
9 brdomg 6960 . . 3  |-  ( B  e.  W  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
1093ad2ant2 977 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
118, 10mpbird 223 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  A  ~<_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 934   E.wex 1541    e. wcel 1710   _Vcvv 2864   class class class wbr 4104   -->wf 5333   -1-1->wf1 5334    ~<_ cdom 6949
This theorem is referenced by:  ssdomg  6995  domdifsn  7033  sucdom2  7145  unxpdomlem3  7157  unbnn  7203  fodomacn  7773  hauspwpwdom  17785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-dom 6953
  Copyright terms: Public domain W3C validator