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Theorem f1dom2g 7489
Description: The domain of a one-to-one function is dominated by its codomain. This variation of f1domg 7491 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 5718 . . . . 5  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 fex2 6691 . . . . 5  |-  ( ( F : A --> B  /\  A  e.  V  /\  B  e.  W )  ->  F  e.  _V )
31, 2syl3an1 1261 . . . 4  |-  ( ( F : A -1-1-> B  /\  A  e.  V  /\  B  e.  W
)  ->  F  e.  _V )
433coml 1202 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F  e.  _V )
5 simp3 997 . . 3  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  F : A -1-1-> B )
6 f1eq1 5713 . . . 4  |-  ( f  =  F  ->  (
f : A -1-1-> B  <->  F : A -1-1-> B ) )
76spcegv 3142 . . 3  |-  ( F  e.  _V  ->  ( F : A -1-1-> B  ->  E. f  f : A -1-1-> B ) )
84, 5, 7sylc 59 . 2  |-  ( ( A  e.  V  /\  B  e.  W  /\  F : A -1-1-> B )  ->  E. f  f : A -1-1-> B )
9 brdomg 7482 . . 3  |-  ( B  e.  W  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
1093ad2ant2 1017 . 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 972   E.wex 1631    e. wcel 1840   _Vcvv 3056   class class class wbr 4392   -->wf 5519   -1-1->wf1 5520    ~<_ cdom 7470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-br 4393  df-opab 4451  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-dom 7474
This theorem is referenced by:  ssdomg  7517  domdifsn  7556  sucdom2  7669  unxpdomlem3  7679  unbnn  7728  fodomacn  8387  hauspwpwdom  20671
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