MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oacomf1olem Structured version   Unicode version

Theorem oacomf1olem 6809
Description: Lemma for oacomf1o 6810. (Contributed by Mario Carneiro, 30-May-2015.)
Hypothesis
Ref Expression
oacomf1olem.1  |-  F  =  ( x  e.  A  |->  ( B  +o  x
) )
Assertion
Ref Expression
oacomf1olem  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( F : A -1-1-onto-> ran  F  /\  ( ran  F  i^i  B )  =  (/) ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    F( x)

Proof of Theorem oacomf1olem
StepHypRef Expression
1 oaf1o 6808 . . . . . . 7  |-  ( B  e.  On  ->  (
x  e.  On  |->  ( B  +o  x ) ) : On -1-1-onto-> ( On  \  B
) )
21adantl 454 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  On  |->  ( B  +o  x
) ) : On -1-1-onto-> ( On  \  B ) )
3 f1of1 5675 . . . . . 6  |-  ( ( x  e.  On  |->  ( B  +o  x ) ) : On -1-1-onto-> ( On  \  B
)  ->  ( x  e.  On  |->  ( B  +o  x ) ) : On -1-1-> ( On  \  B ) )
42, 3syl 16 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  On  |->  ( B  +o  x
) ) : On -1-1-> ( On  \  B ) )
5 onss 4773 . . . . . 6  |-  ( A  e.  On  ->  A  C_  On )
65adantr 453 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  On )
7 f1ssres 5648 . . . . 5  |-  ( ( ( x  e.  On  |->  ( B  +o  x
) ) : On -1-1-> ( On  \  B )  /\  A  C_  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A ) : A -1-1-> ( On  \  B ) )
84, 6, 7syl2anc 644 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A ) : A -1-1-> ( On  \  B ) )
9 resmpt 5193 . . . . . . 7  |-  ( A 
C_  On  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  ( x  e.  A  |->  ( B  +o  x ) ) )
106, 9syl 16 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  ( x  e.  A  |->  ( B  +o  x ) ) )
11 oacomf1olem.1 . . . . . 6  |-  F  =  ( x  e.  A  |->  ( B  +o  x
) )
1210, 11syl6eqr 2488 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  F )
13 f1eq1 5636 . . . . 5  |-  ( ( ( x  e.  On  |->  ( B  +o  x
) )  |`  A )  =  F  ->  (
( ( x  e.  On  |->  ( B  +o  x ) )  |`  A ) : A -1-1-> ( On  \  B )  <-> 
F : A -1-1-> ( On  \  B ) ) )
1412, 13syl 16 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A ) : A -1-1-> ( On  \  B )  <-> 
F : A -1-1-> ( On  \  B ) ) )
158, 14mpbid 203 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : A -1-1-> ( On  \  B ) )
16 f1f1orn 5687 . . 3  |-  ( F : A -1-1-> ( On 
\  B )  ->  F : A -1-1-onto-> ran  F )
1715, 16syl 16 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : A -1-1-onto-> ran  F
)
18 f1f 5641 . . . 4  |-  ( F : A -1-1-> ( On 
\  B )  ->  F : A --> ( On 
\  B ) )
19 frn 5599 . . . 4  |-  ( F : A --> ( On 
\  B )  ->  ran  F  C_  ( On  \  B ) )
2015, 18, 193syl 19 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  ( On  \  B ) )
2120difss2d 3479 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  On )
22 reldisj 3673 . . . 4  |-  ( ran 
F  C_  On  ->  ( ( ran  F  i^i  B )  =  (/)  <->  ran  F  C_  ( On  \  B ) ) )
2321, 22syl 16 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( ran  F  i^i  B )  =  (/)  <->  ran  F 
C_  ( On  \  B ) ) )
2420, 23mpbird 225 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  F  i^i  B )  =  (/) )
2517, 24jca 520 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( F : A -1-1-onto-> ran  F  /\  ( ran  F  i^i  B )  =  (/) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3319    i^i cin 3321    C_ wss 3322   (/)c0 3630    e. cmpt 4268   Oncon0 4583   ran crn 4881    |` cres 4882   -->wf 5452   -1-1->wf1 5453   -1-1-onto->wf1o 5455  (class class class)co 6083    +o coa 6723
This theorem is referenced by:  oacomf1o  6810  onacda  8079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-recs 6635  df-rdg 6670  df-oadd 6730
  Copyright terms: Public domain W3C validator