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Theorem unfilem2 7372
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 6106 . . . . . 6  |-  ( A  +o  x )  e. 
_V
2 unfilem1.3 . . . . . 6  |-  F  =  ( x  e.  B  |->  ( A  +o  x
) )
31, 2fnmpti 5573 . . . . 5  |-  F  Fn  B
4 unfilem1.1 . . . . . 6  |-  A  e. 
om
5 unfilem1.2 . . . . . 6  |-  B  e. 
om
64, 5, 2unfilem1 7371 . . . . 5  |-  ran  F  =  ( ( A  +o  B )  \  A )
7 df-fo 5460 . . . . 5  |-  ( F : B -onto-> ( ( A  +o  B ) 
\  A )  <->  ( F  Fn  B  /\  ran  F  =  ( ( A  +o  B )  \  A ) ) )
83, 6, 7mpbir2an 887 . . . 4  |-  F : B -onto-> ( ( A  +o  B )  \  A )
9 fof 5653 . . . 4  |-  ( F : B -onto-> ( ( A  +o  B ) 
\  A )  ->  F : B --> ( ( A  +o  B ) 
\  A ) )
108, 9ax-mp 8 . . 3  |-  F : B
--> ( ( A  +o  B )  \  A
)
11 oveq2 6089 . . . . . . . 8  |-  ( x  =  z  ->  ( A  +o  x )  =  ( A  +o  z
) )
12 ovex 6106 . . . . . . . 8  |-  ( A  +o  z )  e. 
_V
1311, 2, 12fvmpt 5806 . . . . . . 7  |-  ( z  e.  B  ->  ( F `  z )  =  ( A  +o  z ) )
14 oveq2 6089 . . . . . . . 8  |-  ( x  =  w  ->  ( A  +o  x )  =  ( A  +o  w
) )
15 ovex 6106 . . . . . . . 8  |-  ( A  +o  w )  e. 
_V
1614, 2, 15fvmpt 5806 . . . . . . 7  |-  ( w  e.  B  ->  ( F `  w )  =  ( A  +o  w ) )
1713, 16eqeqan12d 2451 . . . . . 6  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
( A  +o  z
)  =  ( A  +o  w ) ) )
18 elnn 4855 . . . . . . . 8  |-  ( ( z  e.  B  /\  B  e.  om )  ->  z  e.  om )
195, 18mpan2 653 . . . . . . 7  |-  ( z  e.  B  ->  z  e.  om )
20 elnn 4855 . . . . . . . 8  |-  ( ( w  e.  B  /\  B  e.  om )  ->  w  e.  om )
215, 20mpan2 653 . . . . . . 7  |-  ( w  e.  B  ->  w  e.  om )
22 nnacan 6871 . . . . . . . 8  |-  ( ( A  e.  om  /\  z  e.  om  /\  w  e.  om )  ->  (
( A  +o  z
)  =  ( A  +o  w )  <->  z  =  w ) )
234, 22mp3an1 1266 . . . . . . 7  |-  ( ( z  e.  om  /\  w  e.  om )  ->  ( ( A  +o  z )  =  ( A  +o  w )  <-> 
z  =  w ) )
2419, 21, 23syl2an 464 . . . . . 6  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( A  +o  z )  =  ( A  +o  w )  <-> 
z  =  w ) )
2517, 24bitrd 245 . . . . 5  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
z  =  w ) )
2625biimpd 199 . . . 4  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
2726rgen2a 2772 . . 3  |-  A. z  e.  B  A. w  e.  B  ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
28 dff13 6004 . . 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 887 . 2  |-  F : B -1-1-> ( ( A  +o  B )  \  A )
30 df-f1o 5461 . 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 887 1  |-  F : B
-1-1-onto-> ( ( A  +o  B )  \  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    \ cdif 3317    e. cmpt 4266   omcom 4845   ran crn 4879    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   -onto->wfo 5452   -1-1-onto->wf1o 5453   ` cfv 5454  (class class class)co 6081    +o coa 6721
This theorem is referenced by:  unfilem3  7373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-recs 6633  df-rdg 6668  df-oadd 6728
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