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Theorem unfilem2 7821
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 6308 . . . . . 6  |-  ( A  +o  x )  e. 
_V
2 unfilem1.3 . . . . . 6  |-  F  =  ( x  e.  B  |->  ( A  +o  x
) )
31, 2fnmpti 5694 . . . . 5  |-  F  Fn  B
4 unfilem1.1 . . . . . 6  |-  A  e. 
om
5 unfilem1.2 . . . . . 6  |-  B  e. 
om
64, 5, 2unfilem1 7820 . . . . 5  |-  ran  F  =  ( ( A  +o  B )  \  A )
7 df-fo 5577 . . . . 5  |-  ( F : B -onto-> ( ( A  +o  B ) 
\  A )  <->  ( F  Fn  B  /\  ran  F  =  ( ( A  +o  B )  \  A ) ) )
83, 6, 7mpbir2an 923 . . . 4  |-  F : B -onto-> ( ( A  +o  B )  \  A )
9 fof 5780 . . . 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 6288 . . . . . . . 8  |-  ( x  =  z  ->  ( A  +o  x )  =  ( A  +o  z
) )
12 ovex 6308 . . . . . . . 8  |-  ( A  +o  z )  e. 
_V
1311, 2, 12fvmpt 5934 . . . . . . 7  |-  ( z  e.  B  ->  ( F `  z )  =  ( A  +o  z ) )
14 oveq2 6288 . . . . . . . 8  |-  ( x  =  w  ->  ( A  +o  x )  =  ( A  +o  w
) )
15 ovex 6308 . . . . . . . 8  |-  ( A  +o  w )  e. 
_V
1614, 2, 15fvmpt 5934 . . . . . . 7  |-  ( w  e.  B  ->  ( F `  w )  =  ( A  +o  w ) )
1713, 16eqeqan12d 2427 . . . . . 6  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
( A  +o  z
)  =  ( A  +o  w ) ) )
18 elnn 6695 . . . . . . . 8  |-  ( ( z  e.  B  /\  B  e.  om )  ->  z  e.  om )
195, 18mpan2 671 . . . . . . 7  |-  ( z  e.  B  ->  z  e.  om )
20 elnn 6695 . . . . . . . 8  |-  ( ( w  e.  B  /\  B  e.  om )  ->  w  e.  om )
215, 20mpan2 671 . . . . . . 7  |-  ( w  e.  B  ->  w  e.  om )
22 nnacan 7316 . . . . . . . 8  |-  ( ( A  e.  om  /\  z  e.  om  /\  w  e.  om )  ->  (
( A  +o  z
)  =  ( A  +o  w )  <->  z  =  w ) )
234, 22mp3an1 1315 . . . . . . 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 255 . . . . 5  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  <-> 
z  =  w ) )
2625biimpd 209 . . . 4  |-  ( ( z  e.  B  /\  w  e.  B )  ->  ( ( F `  z )  =  ( F `  w )  ->  z  =  w ) )
2726rgen2a 2833 . . 3  |-  A. z  e.  B  A. w  e.  B  ( ( F `  z )  =  ( F `  w )  ->  z  =  w )
28 dff13 6149 . . 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 923 . 2  |-  F : B -1-1-> ( ( A  +o  B )  \  A )
30 df-f1o 5578 . 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 923 1  |-  F : B
-1-1-onto-> ( ( A  +o  B )  \  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    = wceq 1407    e. wcel 1844   A.wral 2756    \ cdif 3413    |-> cmpt 4455   ran crn 4826    Fn wfn 5566   -->wf 5567   -1-1->wf1 5568   -onto->wfo 5569   -1-1-onto->wf1o 5570   ` cfv 5571  (class class class)co 6280   omcom 6685    +o coa 7166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-oadd 7173
This theorem is referenced by:  unfilem3  7822
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