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Theorem djudom 6787
Description: Dominance law for disjoint union. (Contributed by Jim Kingdon, 25-Jul-2022.)
Assertion
Ref Expression
djudom  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( A C )  ~<_  ( B D ) )

Proof of Theorem djudom
Dummy variables  f  g  h are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 brdomi 6466 . . 3  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
21adantr 270 . 2  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  E. f 
f : A -1-1-> B
)
3 brdomi 6466 . . . 4  |-  ( C  ~<_  D  ->  E. g 
g : C -1-1-> D
)
43ad2antlr 473 . . 3  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  ->  E. g  g : C -1-1-> D )
5 inlresf1 6753 . . . . . . . . 9  |-  (inl  |`  B ) : B -1-1-> ( B D )
6 simplr 497 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
f : A -1-1-> B
)
7 f1co 5228 . . . . . . . . 9  |-  ( ( (inl  |`  B ) : B -1-1-> ( B D )  /\  f : A -1-1-> B )  ->  ( (inl  |`  B )  o.  f
) : A -1-1-> ( B D ) )
85, 6, 7sylancr 405 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( (inl  |`  B )  o.  f ) : A -1-1-> ( B D ) )
9 inrresf1 6754 . . . . . . . . 9  |-  (inr  |`  D ) : D -1-1-> ( B D )
10 simpr 108 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
g : C -1-1-> D
)
11 f1co 5228 . . . . . . . . 9  |-  ( ( (inr  |`  D ) : D -1-1-> ( B D )  /\  g : C -1-1-> D )  ->  ( (inr  |`  D )  o.  g
) : C -1-1-> ( B D ) )
129, 10, 11sylancr 405 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( (inr  |`  D )  o.  g ) : C -1-1-> ( B D ) )
13 rnco 4937 . . . . . . . . . . 11  |-  ran  (
(inl  |`  B )  o.  f )  =  ran  ( (inl  |`  B )  |`  ran  f )
14 f1rn 5217 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
1514ad2antlr 473 . . . . . . . . . . . . 13  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  ran  f  C_  B )
16 resabs1 4742 . . . . . . . . . . . . 13  |-  ( ran  f  C_  B  ->  ( (inl  |`  B )  |`  ran  f )  =  (inl  |`  ran  f ) )
1715, 16syl 14 . . . . . . . . . . . 12  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( (inl  |`  B )  |`  ran  f )  =  (inl  |`  ran  f ) )
1817rneqd 4664 . . . . . . . . . . 11  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  ran  ( (inl  |`  B )  |`  ran  f )  =  ran  (inl  |`  ran  f
) )
1913, 18syl5eq 2132 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  ran  ( (inl  |`  B )  o.  f )  =  ran  (inl  |`  ran  f
) )
20 rnco 4937 . . . . . . . . . . 11  |-  ran  (
(inr  |`  D )  o.  g )  =  ran  ( (inr  |`  D )  |`  ran  g )
21 f1rn 5217 . . . . . . . . . . . . 13  |-  ( g : C -1-1-> D  ->  ran  g  C_  D )
22 resabs1 4742 . . . . . . . . . . . . 13  |-  ( ran  g  C_  D  ->  ( (inr  |`  D )  |`  ran  g )  =  (inr  |`  ran  g ) )
2310, 21, 223syl 17 . . . . . . . . . . . 12  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( (inr  |`  D )  |`  ran  g )  =  (inr  |`  ran  g ) )
2423rneqd 4664 . . . . . . . . . . 11  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  ran  ( (inr  |`  D )  |`  ran  g )  =  ran  (inr  |`  ran  g
) )
2520, 24syl5eq 2132 . . . . . . . . . 10  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  ran  ( (inr  |`  D )  o.  g )  =  ran  (inr  |`  ran  g
) )
2619, 25ineq12d 3202 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( ran  ( (inl  |`  B )  o.  f
)  i^i  ran  ( (inr  |`  D )  o.  g
) )  =  ( ran  (inl  |`  ran  f
)  i^i  ran  (inr  |`  ran  g
) ) )
27 djuinr 6755 . . . . . . . . 9  |-  ( ran  (inl  |`  ran  f )  i^i  ran  (inr  |`  ran  g
) )  =  (/)
2826, 27syl6eq 2136 . . . . . . . 8  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( ran  ( (inl  |`  B )  o.  f
)  i^i  ran  ( (inr  |`  D )  o.  g
) )  =  (/) )
298, 12, 28casef1 6781 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) ) : ( A C )
-1-1-> ( B D )
)
30 f1f 5216 . . . . . . 7  |-  (case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) ) : ( A C )
-1-1-> ( B D )  -> case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) ) : ( A C ) --> ( B D )
)
3129, 30syl 14 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) ) : ( A C ) --> ( B D )
)
32 reldom 6462 . . . . . . . . 9  |-  Rel  ~<_
3332brrelexi 4479 . . . . . . . 8  |-  ( A  ~<_  B  ->  A  e.  _V )
3433ad3antrrr 476 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  A  e.  _V )
3532brrelexi 4479 . . . . . . . 8  |-  ( C  ~<_  D  ->  C  e.  _V )
3635ad3antlr 477 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  C  e.  _V )
37 djuex 6736 . . . . . . 7  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A C )  e.  _V )
3834, 36, 37syl2anc 403 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( A C )  e.  _V )
39 fex 5524 . . . . . 6  |-  ( (case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) ) : ( A C ) --> ( B D )  /\  ( A C )  e.  _V )  -> case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) )  e. 
_V )
4031, 38, 39syl2anc 403 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) )  e.  _V )
41 f1eq1 5211 . . . . . 6  |-  ( h  = case ( ( (inl  |`  B )  o.  f
) ,  ( (inr  |`  D )  o.  g
) )  ->  (
h : ( A C ) -1-1-> ( B D )  <-> case ( (
(inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) ) : ( A C ) -1-1-> ( B D )
) )
4241spcegv 2707 . . . . 5  |-  (case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) )  e.  _V  ->  (case ( ( (inl  |`  B )  o.  f ) ,  ( (inr  |`  D )  o.  g ) ) : ( A C )
-1-1-> ( B D )  ->  E. h  h : ( A C ) -1-1-> ( B D )
) )
4340, 29, 42sylc 61 . . . 4  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  E. h  h :
( A C ) -1-1-> ( B D )
)
4432brrelex2i 4482 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
4544ad3antrrr 476 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  B  e.  _V )
4632brrelex2i 4482 . . . . . 6  |-  ( C  ~<_  D  ->  D  e.  _V )
4746ad3antlr 477 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  D  e.  _V )
48 djuex 6736 . . . . . 6  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B D )  e.  _V )
49 brdomg 6465 . . . . . 6  |-  ( ( B D )  e.  _V  ->  ( ( A C )  ~<_  ( B D )  <->  E. h  h : ( A C ) -1-1-> ( B D ) ) )
5048, 49syl 14 . . . . 5  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( ( A C )  ~<_  ( B D )  <->  E. h  h : ( A C ) -1-1-> ( B D ) ) )
5145, 47, 50syl2anc 403 . . . 4  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( ( A C )  ~<_  ( B D )  <->  E. h  h : ( A C ) -1-1-> ( B D ) ) )
5243, 51mpbird 165 . . 3  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( A C )  ~<_  ( B D ) )
534, 52exlimddv 1826 . 2  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  ->  ( A C )  ~<_  ( B D )
)
542, 53exlimddv 1826 1  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( A C )  ~<_  ( B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619    i^i cin 2998    C_ wss 2999   (/)c0 3286   class class class wbr 3845   ran crn 4439    |` cres 4440    o. ccom 4442   -->wf 5011   -1-1->wf1 5012    ~<_ cdom 6456   ⊔ cdju 6730  inlcinl 6737  inrcinr 6738  casecdjucase 6774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-iord 4193  df-on 4195  df-suc 4198  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-1st 5911  df-2nd 5912  df-1o 6181  df-dom 6459  df-dju 6731  df-inl 6739  df-inr 6740  df-case 6775
This theorem is referenced by:  exmidfodomrlemr  6828  exmidfodomrlemrALT  6829
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