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Theorem unfilem2 7781
Description: Lemma for proving that the union of two finite sets is finite. (Contributed by NM, 10-Nov-2002.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
unfilem1.1  |-  A  e. 
om
unfilem1.2  |-  B  e. 
om
unfilem1.3  |-  F  =  ( x  e.  B  |->  ( A  +o  x
) )
Assertion
Ref Expression
unfilem2  |-  F : B
-1-1-onto-> ( ( A  +o  B )  \  A
)
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    F( x)

Proof of Theorem unfilem2
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6307 . . . . . 6  |-  ( A  +o  x )  e. 
_V
2 unfilem1.3 . . . . . 6  |-  F  =  ( x  e.  B  |->  ( A  +o  x
) )
31, 2fnmpti 5707 . . . . 5  |-  F  Fn  B
4 unfilem1.1 . . . . . 6  |-  A  e. 
om
5 unfilem1.2 . . . . . 6  |-  B  e. 
om
64, 5, 2unfilem1 7780 . . . . 5  |-  ran  F  =  ( ( A  +o  B )  \  A )
7 df-fo 5592 . . . . 5  |-  ( F : B -onto-> ( ( A  +o  B ) 
\  A )  <->  ( F  Fn  B  /\  ran  F  =  ( ( A  +o  B )  \  A ) ) )
83, 6, 7mpbir2an 918 . . . 4  |-  F : B -onto-> ( ( A  +o  B )  \  A )
9 fof 5793 . . . 4  |-  ( F : B -onto-> ( ( A  +o  B ) 
\  A )  ->  F : B --> ( ( A  +o  B ) 
\  A ) )
108, 9ax-mp 5 . . 3  |-  F : B
--> ( ( A  +o  B )  \  A
)
11 oveq2 6290 . . . . . . . 8  |-  ( x  =  z  ->  ( A  +o  x )  =  ( A  +o  z
) )
12 ovex 6307 . . . . . . . 8  |-  ( A  +o  z )  e. 
_V
1311, 2, 12fvmpt 5948 . . . . . . 7  |-  ( z  e.  B  ->  ( F `  z )  =  ( A  +o  z ) )
14 oveq2 6290 . . . . . . . 8  |-  ( x  =  w  ->  ( A  +o  x )  =  ( A  +o  w
) )
15 ovex 6307 . . . . . . . 8  |-  ( A  +o  w )  e. 
_V
1614, 2, 15fvmpt 5948 . . . . . . 7  |-  ( w  e.  B  ->  ( F `  w )  =  ( A  +o  w ) )
1713, 16eqeqan12d 2490 . . . . . 6  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
( A  +o  z
)  =  ( A  +o  w ) ) )
18 elnn 6688 . . . . . . . 8  |-  ( ( z  e.  B  /\  B  e.  om )  ->  z  e.  om )
195, 18mpan2 671 . . . . . . 7  |-  ( z  e.  B  ->  z  e.  om )
20 elnn 6688 . . . . . . . 8  |-  ( ( w  e.  B  /\  B  e.  om )  ->  w  e.  om )
215, 20mpan2 671 . . . . . . 7  |-  ( w  e.  B  ->  w  e.  om )
22 nnacan 7274 . . . . . . . 8  |-  ( ( A  e.  om  /\  z  e.  om  /\  w  e.  om )  ->  (
( A  +o  z
)  =  ( A  +o  w )  <->  z  =  w ) )
234, 22mp3an1 1311 . . . . . . 7  |-  ( ( z  e.  om  /\  w  e.  om )  ->  ( ( A  +o  z )  =  ( A  +o  w )  <-> 
z  =  w ) )
2419, 21, 23syl2an 477 . . . . . 6  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( A  +o  z )  =  ( A  +o  w )  <-> 
z  =  w ) )
2517, 24bitrd 253 . . . . 5  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
z  =  w ) )
2625biimpd 207 . . . 4  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
2726rgen2a 2891 . . 3  |-  A. z  e.  B  A. w  e.  B  ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
28 dff13 6152 . . 3  |-  ( F : B -1-1-> ( ( A  +o  B ) 
\  A )  <->  ( F : B --> ( ( A  +o  B )  \  A )  /\  A. z  e.  B  A. w  e.  B  (
( F `  z
)  =  ( F `
 w )  -> 
z  =  w ) ) )
2910, 27, 28mpbir2an 918 . 2  |-  F : B -1-1-> ( ( A  +o  B )  \  A )
30 df-f1o 5593 . 2  |-  ( F : B -1-1-onto-> ( ( A  +o  B )  \  A
)  <->  ( F : B -1-1-> ( ( A  +o  B )  \  A )  /\  F : B -onto-> ( ( A  +o  B )  \  A ) ) )
3129, 8, 30mpbir2an 918 1  |-  F : B
-1-1-onto-> ( ( A  +o  B )  \  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    \ cdif 3473    |-> cmpt 4505   ran crn 5000    Fn wfn 5581   -->wf 5582   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   omcom 6678    +o coa 7124
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-oadd 7131
This theorem is referenced by:  unfilem3  7782
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