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Theorem unfilem2 7122
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 5883 . . . . . 6  |-  ( A  +o  x )  e. 
_V
2 unfilem1.3 . . . . . 6  |-  F  =  ( x  e.  B  |->  ( A  +o  x
) )
31, 2fnmpti 5372 . . . . 5  |-  F  Fn  B
4 unfilem1.1 . . . . . 6  |-  A  e. 
om
5 unfilem1.2 . . . . . 6  |-  B  e. 
om
64, 5, 2unfilem1 7121 . . . . 5  |-  ran  F  =  ( ( A  +o  B )  \  A )
7 df-fo 5261 . . . . 5  |-  ( F : B -onto-> ( ( A  +o  B ) 
\  A )  <->  ( F  Fn  B  /\  ran  F  =  ( ( A  +o  B )  \  A ) ) )
83, 6, 7mpbir2an 886 . . . 4  |-  F : B -onto-> ( ( A  +o  B )  \  A )
9 fof 5451 . . . 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 5866 . . . . . . . 8  |-  ( x  =  z  ->  ( A  +o  x )  =  ( A  +o  z
) )
12 ovex 5883 . . . . . . . 8  |-  ( A  +o  z )  e. 
_V
1311, 2, 12fvmpt 5602 . . . . . . 7  |-  ( z  e.  B  ->  ( F `  z )  =  ( A  +o  z ) )
14 oveq2 5866 . . . . . . . 8  |-  ( x  =  w  ->  ( A  +o  x )  =  ( A  +o  w
) )
15 ovex 5883 . . . . . . . 8  |-  ( A  +o  w )  e. 
_V
1614, 2, 15fvmpt 5602 . . . . . . 7  |-  ( w  e.  B  ->  ( F `  w )  =  ( A  +o  w ) )
1713, 16eqeqan12d 2298 . . . . . 6  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
( A  +o  z
)  =  ( A  +o  w ) ) )
18 elnn 4666 . . . . . . . 8  |-  ( ( z  e.  B  /\  B  e.  om )  ->  z  e.  om )
195, 18mpan2 652 . . . . . . 7  |-  ( z  e.  B  ->  z  e.  om )
20 elnn 4666 . . . . . . . 8  |-  ( ( w  e.  B  /\  B  e.  om )  ->  w  e.  om )
215, 20mpan2 652 . . . . . . 7  |-  ( w  e.  B  ->  w  e.  om )
22 nnacan 6626 . . . . . . . 8  |-  ( ( A  e.  om  /\  z  e.  om  /\  w  e.  om )  ->  (
( A  +o  z
)  =  ( A  +o  w )  <->  z  =  w ) )
234, 22mp3an1 1264 . . . . . . 7  |-  ( ( z  e.  om  /\  w  e.  om )  ->  ( ( A  +o  z )  =  ( A  +o  w )  <-> 
z  =  w ) )
2419, 21, 23syl2an 463 . . . . . 6  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( A  +o  z )  =  ( A  +o  w )  <-> 
z  =  w ) )
2517, 24bitrd 244 . . . . 5  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
z  =  w ) )
2625biimpd 198 . . . 4  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
2726rgen2a 2609 . . 3  |-  A. z  e.  B  A. w  e.  B  ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
28 dff13 5783 . . 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 886 . 2  |-  F : B -1-1-> ( ( A  +o  B )  \  A )
30 df-f1o 5262 . 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 886 1  |-  F : B
-1-1-onto-> ( ( A  +o  B )  \  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    e. cmpt 4077   omcom 4656   ran crn 4690    Fn wfn 5250   -->wf 5251   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858    +o coa 6476
This theorem is referenced by:  unfilem3  7123
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483
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