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Theorem djudom 7032
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 6691 . . 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 6691 . . . 4  |-  ( C  ~<_  D  ->  E. g 
g : C -1-1-> D
)
43ad2antlr 481 . . 3  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  ->  E. g  g : C -1-1-> D )
5 inlresf1 7000 . . . . . . . . 9  |-  (inl  |`  B ) : B -1-1-> ( B D )
6 simplr 520 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
f : A -1-1-> B
)
7 f1co 5386 . . . . . . . . 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 411 . . . . . . . 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 7001 . . . . . . . . 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 5386 . . . . . . . . 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 411 . . . . . . . 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 5091 . . . . . . . . . . 11  |-  ran  (
(inl  |`  B )  o.  f )  =  ran  ( (inl  |`  B )  |`  ran  f )
14 f1rn 5375 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
1514ad2antlr 481 . . . . . . . . . . . . 13  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  ran  f  C_  B )
16 resabs1 4894 . . . . . . . . . . . . 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 4814 . . . . . . . . . . 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 2202 . . . . . . . . . 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 5091 . . . . . . . . . . 11  |-  ran  (
(inr  |`  D )  o.  g )  =  ran  ( (inr  |`  D )  |`  ran  g )
21 f1rn 5375 . . . . . . . . . . . . 13  |-  ( g : C -1-1-> D  ->  ran  g  C_  D )
22 resabs1 4894 . . . . . . . . . . . . 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 4814 . . . . . . . . . . 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 2202 . . . . . . . . . 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 3309 . . . . . . . . 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 7002 . . . . . . . . 9  |-  ( ran  (inl  |`  ran  f )  i^i  ran  (inr  |`  ran  g
) )  =  (/)
2826, 27eqtrdi 2206 . . . . . . . 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 7029 . . . . . . 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 5374 . . . . . . 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 6687 . . . . . . . . 9  |-  Rel  ~<_
3332brrelex1i 4628 . . . . . . . 8  |-  ( A  ~<_  B  ->  A  e.  _V )
3433ad3antrrr 484 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  A  e.  _V )
3532brrelex1i 4628 . . . . . . . 8  |-  ( C  ~<_  D  ->  C  e.  _V )
3635ad3antlr 485 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  C  e.  _V )
37 djuex 6982 . . . . . . 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 5693 . . . . . 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 5369 . . . . . 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 2800 . . . . 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 4629 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
4544ad3antrrr 484 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  B  e.  _V )
4632brrelex2i 4629 . . . . . 6  |-  ( C  ~<_  D  ->  D  e.  _V )
4746ad3antlr 485 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  D  e.  _V )
48 djuex 6982 . . . . . 6  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B D )  e.  _V )
49 brdomg 6690 . . . . . 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 1878 . 2  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  ->  ( A C )  ~<_  ( B D )
)
542, 53exlimddv 1878 1  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( A C )  ~<_  ( B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   _Vcvv 2712    i^i cin 3101    C_ wss 3102   (/)c0 3394   class class class wbr 3965   ran crn 4586    |` cres 4587    o. ccom 4589   -->wf 5165   -1-1->wf1 5166    ~<_ cdom 6681   ⊔ cdju 6976  inlcinl 6984  inrcinr 6985  casecdjucase 7022
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-1st 6085  df-2nd 6086  df-1o 6360  df-dom 6684  df-dju 6977  df-inl 6986  df-inr 6987  df-case 7023
This theorem is referenced by:  exmidfodomrlemr  7132  exmidfodomrlemrALT  7133  sbthom  13584
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