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Theorem f1dom2g 6730
Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 6732 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 5401 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 5364 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1266 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
433coml 1205 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F  e.  _V )
5 simp3 994 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
6 f1eq1 5396 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-> B  <->  F : A -1-1-> B ) )
76spcegv 2818 . . 3  |-  ( F  e.  _V  ->  ( F : A -1-1-> B  ->  E. f  f : A -1-1-> B ) )
84, 5, 7sylc 62 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  E. f  f : A -1-1-> B )
9 brdomg 6722 . . 3  |-  ( B  e.  W  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
1093ad2ant2 1014 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
118, 10mpbird 166 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 104    /\ w3a 973   E.wex 1485    e. wcel 2141   _Vcvv 2730   class class class wbr 3987   -->wf 5192   -1-1->wf1 5193    ~<_ cdom 6713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-dom 6716
This theorem is referenced by:  ssdomg  6752
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