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Theorem f1dom2g 7440
Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 7442 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 5717 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 6645 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1252 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
433coml 1195 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F  e.  _V )
5 simp3 990 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
6 f1eq1 5712 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-> B  <->  F : A -1-1-> B ) )
76spcegv 3164 . . 3  |-  ( F  e.  _V  ->  ( F : A -1-1-> B  ->  E. f  f : A -1-1-> B ) )
84, 5, 7sylc 60 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  E. f  f : A -1-1-> B )
9 brdomg 7433 . . 3  |-  ( B  e.  W  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
1093ad2ant2 1010 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  ( A  ~<_  B  <->  E. f  f : A -1-1-> B ) )
118, 10mpbird 232 1  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  A  ~<_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965   E.wex 1587    e. wcel 1758   _Vcvv 3078   class class class wbr 4403   -->wf 5525   -1-1->wf1 5526    ~<_ cdom 7421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-dom 7425
This theorem is referenced by:  ssdomg  7468  domdifsn  7507  sucdom2  7621  unxpdomlem3  7633  unbnn  7682  fodomacn  8340  hauspwpwdom  19696
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