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Theorem djudom 6971
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 6636 . . 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 6636 . . . 4  |-  ( C  ~<_  D  ->  E. g 
g : C -1-1-> D
)
43ad2antlr 480 . . 3  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  ->  E. g  g : C -1-1-> D )
5 inlresf1 6939 . . . . . . . . 9  |-  (inl  |`  B ) : B -1-1-> ( B D )
6 simplr 519 . . . . . . . . 9  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
f : A -1-1-> B
)
7 f1co 5335 . . . . . . . . 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 410 . . . . . . . 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 6940 . . . . . . . . 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 5335 . . . . . . . . 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 410 . . . . . . . 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 5040 . . . . . . . . . . 11  |-  ran  (
(inl  |`  B )  o.  f )  =  ran  ( (inl  |`  B )  |`  ran  f )
14 f1rn 5324 . . . . . . . . . . . . . 14  |-  ( f : A -1-1-> B  ->  ran  f  C_  B )
1514ad2antlr 480 . . . . . . . . . . . . 13  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  ran  f  C_  B )
16 resabs1 4843 . . . . . . . . . . . . 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 4763 . . . . . . . . . . 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 2182 . . . . . . . . . 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 5040 . . . . . . . . . . 11  |-  ran  (
(inr  |`  D )  o.  g )  =  ran  ( (inr  |`  D )  |`  ran  g )
21 f1rn 5324 . . . . . . . . . . . . 13  |-  ( g : C -1-1-> D  ->  ran  g  C_  D )
22 resabs1 4843 . . . . . . . . . . . . 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 4763 . . . . . . . . . . 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 2182 . . . . . . . . . 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 3273 . . . . . . . . 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 6941 . . . . . . . . 9  |-  ( ran  (inl  |`  ran  f )  i^i  ran  (inr  |`  ran  g
) )  =  (/)
2826, 27syl6eq 2186 . . . . . . . 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 6968 . . . . . . 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 5323 . . . . . . 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 6632 . . . . . . . . 9  |-  Rel  ~<_
3332brrelex1i 4577 . . . . . . . 8  |-  ( A  ~<_  B  ->  A  e.  _V )
3433ad3antrrr 483 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  A  e.  _V )
3532brrelex1i 4577 . . . . . . . 8  |-  ( C  ~<_  D  ->  C  e.  _V )
3635ad3antlr 484 . . . . . . 7  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  C  e.  _V )
37 djuex 6921 . . . . . . 7  |-  ( ( A  e.  _V  /\  C  e.  _V )  ->  ( A C )  e.  _V )
3834, 36, 37syl2anc 408 . . . . . 6  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  -> 
( A C )  e.  _V )
39 fex 5640 . . . . . 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 408 . . . . 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 5318 . . . . . 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 2769 . . . . 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 4578 . . . . . 6  |-  ( A  ~<_  B  ->  B  e.  _V )
4544ad3antrrr 483 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  B  e.  _V )
4632brrelex2i 4578 . . . . . 6  |-  ( C  ~<_  D  ->  D  e.  _V )
4746ad3antlr 484 . . . . 5  |-  ( ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  /\  g : C -1-1-> D )  ->  D  e.  _V )
48 djuex 6921 . . . . . 6  |-  ( ( B  e.  _V  /\  D  e.  _V )  ->  ( B D )  e.  _V )
49 brdomg 6635 . . . . . 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 408 . . . 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 1870 . 2  |-  ( ( ( A  ~<_  B  /\  C  ~<_  D )  /\  f : A -1-1-> B )  ->  ( A C )  ~<_  ( B D )
)
542, 53exlimddv 1870 1  |-  ( ( A  ~<_  B  /\  C  ~<_  D )  ->  ( A C )  ~<_  ( B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1331   E.wex 1468    e. wcel 1480   _Vcvv 2681    i^i cin 3065    C_ wss 3066   (/)c0 3358   class class class wbr 3924   ran crn 4535    |` cres 4536    o. ccom 4538   -->wf 5114   -1-1->wf1 5115    ~<_ cdom 6626   ⊔ cdju 6915  inlcinl 6923  inrcinr 6924  casecdjucase 6961
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1st 6031  df-2nd 6032  df-1o 6306  df-dom 6629  df-dju 6916  df-inl 6925  df-inr 6926  df-case 6962
This theorem is referenced by:  exmidfodomrlemr  7051  exmidfodomrlemrALT  7052  sbthom  13210
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