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Theorem oacomf1olem 6562
Description: Lemma for oacomf1o 6563. (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 6561 . . . . . . 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 5471 . . . . . 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 4582 . . . . . 6  |-  ( A  e.  On  ->  A  C_  On )
65adantr 451 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  On )
7 f1ssres 5444 . . . . 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 5000 . . . . . . 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 2333 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( x  e.  On  |->  ( B  +o  x ) )  |`  A )  =  F )
13 f1eq1 5432 . . . . 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 5483 . . 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 5437 . . . 4  |-  ( F : A -1-1-> ( On 
\  B )  ->  F : A --> ( On 
\  B ) )
19 frn 5395 . . . 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 3303 . . . . 5  |-  ( On 
\  B )  C_  On
2220, 21syl6ss 3191 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ran  F  C_  On )
23 reldisj 3498 . . . 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 1623    e. wcel 1684    \ cdif 3149    i^i cin 3151    C_ wss 3152   (/)c0 3455    e. cmpt 4077   Oncon0 4392   ran crn 4690    |` cres 4691   -->wf 5251   -1-1->wf1 5252   -1-1-onto->wf1o 5254  (class class class)co 5858    +o coa 6476
This theorem is referenced by:  oacomf1o  6563  onacda  7823
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483
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