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Theorem djudom 7070
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 6727 . . 3  |-  ( A  ~<_  B  ->  E. f 
f : A -1-1-> B
)
21adantr 274 . 2  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  E. f 
f : A -1-1-> B
)
3 brdomi 6727 . . . 4  |-  ( C  ~<_  D  ->  E. g 
g : C -1-1-> D
)
43ad2antlr 486 . . 3  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  ->  E. g  g : C -1-1-> D )
5 inlresf1 7038 . . . . . . . . 9  |-  (inl  |`  B ) : B -1-1-> ( B D )
6 simplr 525 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
f : A -1-1-> B
)
7 f1co 5415 . . . . . . . . 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 412 . . . . . . . 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 7039 . . . . . . . . 9  |-  (inr  |`  D ) : D -1-1-> ( B D )
10 simpr 109 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
g : C -1-1-> D
)
11 f1co 5415 . . . . . . . . 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 412 . . . . . . . 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 5117 . . . . . . . . . . 11  |-  ran  (
(inl  |`  B )  o.  f )  =  ran  ( (inl  |`  B )  |`  ran  f )
14 f1rn 5404 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
1514ad2antlr 486 . . . . . . . . . . . . 13  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  ran  f  C_  B )
16 resabs1 4920 . . . . . . . . . . . . 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 4840 . . . . . . . . . . 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, 18eqtrid 2215 . . . . . . . . . 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 5117 . . . . . . . . . . 11  |-  ran  (
(inr  |`  D )  o.  g )  =  ran  ( (inr  |`  D )  |`  ran  g )
21 f1rn 5404 . . . . . . . . . . . . 13  |-  ( g : C -1-1-> D  ->  ran  g  C_  D )
22 resabs1 4920 . . . . . . . . . . . . 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 4840 . . . . . . . . . . 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, 24eqtrid 2215 . . . . . . . . . 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 3329 . . . . . . . . 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 7040 . . . . . . . . 9  |-  ( ran  (inl  |`  ran  f )  i^i  ran  (inr  |`  ran  g
) )  =  (/)
2826, 27eqtrdi 2219 . . . . . . . 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 7067 . . . . . . 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 5403 . . . . . . 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 6723 . . . . . . . . 9  |-  Rel  ~<_
3332brrelex1i 4654 . . . . . . . 8  |-  ( A  ~<_  B  ->  A  e.  _V )
3433ad3antrrr 489 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  A  e.  _V )
3532brrelex1i 4654 . . . . . . . 8  |-  ( C  ~<_  D  ->  C  e.  _V )
3635ad3antlr 490 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  C  e.  _V )
37 djuex 7020 . . . . . . 7  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A C )  e.  _V )
3834, 36, 37syl2anc 409 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( A C )  e.  _V )
39 fex 5725 . . . . . 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 409 . . . . 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 5398 . . . . . 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 2818 . . . . 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 62 . . . 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 4655 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
4544ad3antrrr 489 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  B  e.  _V )
4632brrelex2i 4655 . . . . . 6  |-  ( C  ~<_  D  ->  D  e.  _V )
4746ad3antlr 490 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  D  e.  _V )
48 djuex 7020 . . . . . 6  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B D )  e.  _V )
49 brdomg 6726 . . . . . 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 409 . . . 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 166 . . 3  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( A C )  ~<_  ( B D ) )
534, 52exlimddv 1891 . 2  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  ->  ( A C )  ~<_  ( B D )
)
542, 53exlimddv 1891 1  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( A C )  ~<_  ( B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1348   E.wex 1485    e. wcel 2141   _Vcvv 2730    i^i cin 3120    C_ wss 3121   (/)c0 3414   class class class wbr 3989   ran crn 4612    |` cres 4613    o. ccom 4615   -->wf 5194   -1-1->wf1 5195    ~<_ cdom 6717   ⊔ cdju 7014  inlcinl 7022  inrcinr 7023  casecdjucase 7060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-1o 6395  df-dom 6720  df-dju 7015  df-inl 7024  df-inr 7025  df-case 7061
This theorem is referenced by:  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  sbthom  14058
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