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Theorem oacomf1olem 7168
Description: Lemma for oacomf1o 7169. (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 7167 . . . . . . 7  |-  ( B  e.  On  ->  (
x  e.  On  |->  ( B  +o  x ) ) : On -1-1-onto-> ( On  \  B
) )
21adantl 464 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  On  |->  ( B  +o  x
) ) : On -1-1-onto-> ( On  \  B ) )
3 f1of1 5752 . . . . . 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 17 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  On  |->  ( B  +o  x
) ) : On -1-1-> ( On  \  B ) )
5 onss 6562 . . . . . 6  |-  ( A  e.  On  ->  A  C_  On )
65adantr 463 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  On )
7 f1ssres 5725 . . . . 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 659 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A ) : A -1-1-> ( On  \  B ) )
96resmptd 5264 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  ( x  e.  A  |->  ( B  +o  x ) ) )
10 oacomf1olem.1 . . . . . 6  |-  F  =  ( x  e.  A  |->  ( B  +o  x
) )
119, 10syl6eqr 2459 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  F )
12 f1eq1 5713 . . . . 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 ) ) )
1311, 12syl 17 . . . 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 ) ) )
148, 13mpbid 210 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : A -1-1-> ( On  \  B ) )
15 f1f1orn 5764 . . 3  |-  ( F : A -1-1-> ( On 
\  B )  ->  F : A -1-1-onto-> ran  F )
1614, 15syl 17 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : A -1-1-onto-> ran  F
)
17 f1f 5718 . . . 4  |-  ( F : A -1-1-> ( On 
\  B )  ->  F : A --> ( On 
\  B ) )
18 frn 5674 . . . 4  |-  ( F : A --> ( On 
\  B )  ->  ran  F  C_  ( On  \  B ) )
1914, 17, 183syl 20 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  ( On  \  B ) )
2019difss2d 3570 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  On )
21 reldisj 3810 . . . 4  |-  ( ran 
F  C_  On  ->  ( ( ran  F  i^i  B )  =  (/)  <->  ran  F  C_  ( On  \  B ) ) )
2220, 21syl 17 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( ran  F  i^i  B )  =  (/)  <->  ran  F 
C_  ( On  \  B ) ) )
2319, 22mpbird 232 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  F  i^i  B )  =  (/) )
2416, 23jca 530 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 367    = wceq 1403    e. wcel 1840    \ cdif 3408    i^i cin 3410    C_ wss 3411   (/)c0 3735    |-> cmpt 4450   Oncon0 4819   ran crn 4941    |` cres 4942   -->wf 5519   -1-1->wf1 5520   -1-1-onto->wf1o 5522  (class class class)co 6232    +o coa 7082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-recs 6997  df-rdg 7031  df-oadd 7089
This theorem is referenced by:  oacomf1o  7169  onacda  8527
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