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Theorem oacomf1olem 6578
Description: Lemma for oacomf1o 6579. (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 6577 . . . . . . 7  |-  ( B  e.  On  ->  (
x  e.  On  |->  ( B  +o  x ) ) : On -1-1-onto-> ( On  \  B
) )
21adantl 452 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  On  |->  ( B  +o  x
) ) : On -1-1-onto-> ( On  \  B ) )
3 f1of1 5487 . . . . . 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 15 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( x  e.  On  |->  ( B  +o  x
) ) : On -1-1-> ( On  \  B ) )
5 onss 4598 . . . . . 6  |-  ( A  e.  On  ->  A  C_  On )
65adantr 451 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  On )
7 f1ssres 5460 . . . . 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 642 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A ) : A -1-1-> ( On  \  B ) )
9 resmpt 5016 . . . . . . 7  |-  ( A 
C_  On  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  ( x  e.  A  |->  ( B  +o  x ) ) )
106, 9syl 15 . . . . . 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 2346 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  F )
13 f1eq1 5448 . . . . 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 15 . . . 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 201 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : A -1-1-> ( On  \  B ) )
16 f1f1orn 5499 . . 3  |-  ( F : A -1-1-> ( On 
\  B )  ->  F : A -1-1-onto-> ran  F )
1715, 16syl 15 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  F : A -1-1-onto-> ran  F
)
18 f1f 5453 . . . 4  |-  ( F : A -1-1-> ( On 
\  B )  ->  F : A --> ( On 
\  B ) )
19 frn 5411 . . . 4  |-  ( F : A --> ( On 
\  B )  ->  ran  F  C_  ( On  \  B ) )
2015, 18, 193syl 18 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  ( On  \  B ) )
21 difss 3316 . . . . 5  |-  ( On 
\  B )  C_  On
2220, 21syl6ss 3204 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  On )
23 reldisj 3511 . . . 4  |-  ( ran 
F  C_  On  ->  ( ( ran  F  i^i  B )  =  (/)  <->  ran  F  C_  ( On  \  B ) ) )
2422, 23syl 15 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( ran  F  i^i  B )  =  (/)  <->  ran  F 
C_  ( On  \  B ) ) )
2520, 24mpbird 223 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ran  F  i^i  B )  =  (/) )
2617, 25jca 518 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 176    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162    i^i cin 3164    C_ wss 3165   (/)c0 3468    e. cmpt 4093   Oncon0 4408   ran crn 4706    |` cres 4707   -->wf 5267   -1-1->wf1 5268   -1-1-onto->wf1o 5270  (class class class)co 5874    +o coa 6492
This theorem is referenced by:  oacomf1o  6579  onacda  7839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-oadd 6499
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