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Theorem unfilem2 7577
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 6116 . . . . . 6  |-  ( A  +o  x )  e. 
_V
2 unfilem1.3 . . . . . 6  |-  F  =  ( x  e.  B  |->  ( A  +o  x
) )
31, 2fnmpti 5539 . . . . 5  |-  F  Fn  B
4 unfilem1.1 . . . . . 6  |-  A  e. 
om
5 unfilem1.2 . . . . . 6  |-  B  e. 
om
64, 5, 2unfilem1 7576 . . . . 5  |-  ran  F  =  ( ( A  +o  B )  \  A )
7 df-fo 5424 . . . . 5  |-  ( F : B -onto-> ( ( A  +o  B ) 
\  A )  <->  ( F  Fn  B  /\  ran  F  =  ( ( A  +o  B )  \  A ) ) )
83, 6, 7mpbir2an 911 . . . 4  |-  F : B -onto-> ( ( A  +o  B )  \  A )
9 fof 5620 . . . 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 6099 . . . . . . . 8  |-  ( x  =  z  ->  ( A  +o  x )  =  ( A  +o  z
) )
12 ovex 6116 . . . . . . . 8  |-  ( A  +o  z )  e. 
_V
1311, 2, 12fvmpt 5774 . . . . . . 7  |-  ( z  e.  B  ->  ( F `  z )  =  ( A  +o  z ) )
14 oveq2 6099 . . . . . . . 8  |-  ( x  =  w  ->  ( A  +o  x )  =  ( A  +o  w
) )
15 ovex 6116 . . . . . . . 8  |-  ( A  +o  w )  e. 
_V
1614, 2, 15fvmpt 5774 . . . . . . 7  |-  ( w  e.  B  ->  ( F `  w )  =  ( A  +o  w ) )
1713, 16eqeqan12d 2458 . . . . . 6  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
( A  +o  z
)  =  ( A  +o  w ) ) )
18 elnn 6486 . . . . . . . 8  |-  ( ( z  e.  B  /\  B  e.  om )  ->  z  e.  om )
195, 18mpan2 671 . . . . . . 7  |-  ( z  e.  B  ->  z  e.  om )
20 elnn 6486 . . . . . . . 8  |-  ( ( w  e.  B  /\  B  e.  om )  ->  w  e.  om )
215, 20mpan2 671 . . . . . . 7  |-  ( w  e.  B  ->  w  e.  om )
22 nnacan 7067 . . . . . . . 8  |-  ( ( A  e.  om  /\  z  e.  om  /\  w  e.  om )  ->  (
( A  +o  z
)  =  ( A  +o  w )  <->  z  =  w ) )
234, 22mp3an1 1301 . . . . . . 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 2782 . . 3  |-  A. z  e.  B  A. w  e.  B  ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
28 dff13 5971 . . 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 911 . 2  |-  F : B -1-1-> ( ( A  +o  B )  \  A )
30 df-f1o 5425 . 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 911 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 1369    e. wcel 1756   A.wral 2715    \ cdif 3325    e. cmpt 4350   ran crn 4841    Fn wfn 5413   -->wf 5414   -1-1->wf1 5415   -onto->wfo 5416   -1-1-onto->wf1o 5417   ` cfv 5418  (class class class)co 6091   omcom 6476    +o coa 6917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-recs 6832  df-rdg 6866  df-oadd 6924
This theorem is referenced by:  unfilem3  7578
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