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Theorem djudoml 7196
Description: A set is dominated by its disjoint union with another. (Contributed by Jim Kingdon, 11-Jul-2023.)
Assertion
Ref Expression
djudoml  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  ( A B ) )

Proof of Theorem djudoml
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 df-inl 7024 . . . . 5  |- inl  =  ( x  e.  _V  |->  <. (/)
,  x >. )
21funmpt2 5237 . . . 4  |-  Fun inl
3 simpl 108 . . . 4  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
4 resfunexg 5717 . . . 4  |-  ( ( Fun inl  /\  A  e.  V )  ->  (inl  |`  A )  e.  _V )
52, 3, 4sylancr 412 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  (inl  |`  A )  e. 
_V )
6 inlresf1 7038 . . 3  |-  (inl  |`  A ) : A -1-1-> ( A B )
7 f1eq1 5398 . . . 4  |-  ( f  =  (inl  |`  A )  ->  ( f : A -1-1-> ( A B )  <-> 
(inl  |`  A ) : A -1-1-> ( A B ) ) )
87spcegv 2818 . . 3  |-  ( (inl  |`  A )  e.  _V  ->  ( (inl  |`  A ) : A -1-1-> ( A B )  ->  E. f 
f : A -1-1-> ( A B ) ) )
95, 6, 8mpisyl 1439 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  E. f  f : A -1-1-> ( A B ) )
10 djuex 7020 . . 3  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A B )  e.  _V )
11 brdomg 6726 . . 3  |-  ( ( A B )  e.  _V  ->  ( A  ~<_  ( A B )  <->  E. f 
f : A -1-1-> ( A B ) ) )
1210, 11syl 14 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  ~<_  ( A B )  <->  E. f 
f : A -1-1-> ( A B ) ) )
139, 12mpbird 166 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  ~<_  ( A B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1485    e. wcel 2141   _Vcvv 2730   (/)c0 3414   <.cop 3586   class class class wbr 3989    |` cres 4613   Fun wfun 5192   -1-1->wf1 5195    ~<_ cdom 6717   ⊔ cdju 7014  inlcinl 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 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-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  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
This theorem is referenced by: (None)
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