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Theorem oacomf1olem 7008
Description: Lemma for oacomf1o 7009. (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 7007 . . . . . . 7  |-  ( B  e.  On  ->  (
x  e.  On  |->  ( B  +o  x ) ) : On -1-1-onto-> ( On  \  B
) )
21adantl 466 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  On  |->  ( B  +o  x
) ) : On -1-1-onto-> ( On  \  B ) )
3 f1of1 5645 . . . . . 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 6407 . . . . . 6  |-  ( A  e.  On  ->  A  C_  On )
65adantr 465 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  On )
7 f1ssres 5618 . . . . 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 661 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A ) : A -1-1-> ( On  \  B ) )
9 resmpt 5161 . . . . . . 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 2493 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  F )
13 f1eq1 5606 . . . . 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 210 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : A -1-1-> ( On  \  B ) )
16 f1f1orn 5657 . . 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 5611 . . . 4  |-  ( F : A -1-1-> ( On 
\  B )  ->  F : A --> ( On 
\  B ) )
19 frn 5570 . . . 4  |-  ( F : A --> ( On 
\  B )  ->  ran  F  C_  ( On  \  B ) )
2015, 18, 193syl 20 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  ( On  \  B ) )
2120difss2d 3491 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  On )
22 reldisj 3727 . . . 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 232 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  F  i^i  B )  =  (/) )
2517, 24jca 532 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( F : A -1-1-onto-> ran  F  /\  ( ran  F  i^i  B )  =  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    \ cdif 3330    i^i cin 3332    C_ wss 3333   (/)c0 3642    e. cmpt 4355   Oncon0 4724   ran crn 4846    |` cres 4847   -->wf 5419   -1-1->wf1 5420   -1-1-onto->wf1o 5422  (class class class)co 6096    +o coa 6922
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-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-oadd 6929
This theorem is referenced by:  oacomf1o  7009  onacda  8371
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