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Theorem ubpar 25261
Description: If  U is an upper bound of  A and  B 
C_  A then  U is an upper bound of  B. Bourbaki E.III.9 n 8. (Contributed by FL, 23-May-2011.)
Assertion
Ref Expression
ubpar  |-  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  ( U  e.  ( R  ub  A )  ->  U  e.  ( R  ub  B
) ) )

Proof of Theorem ubpar
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2283 . . . . 5  |-  dom  R  =  dom  R
21puub2 25258 . . . 4  |-  ( ( R  e. PresetRel  /\  A  e.  C )  ->  ( U  e.  ( R  ub  A )  <->  ( U  e.  dom  R  /\  A. x  e.  A  x R U ) ) )
32biimpd 198 . . 3  |-  ( ( R  e. PresetRel  /\  A  e.  C )  ->  ( U  e.  ( R  ub  A )  ->  ( U  e.  dom  R  /\  A. x  e.  A  x R U ) ) )
433adant3 975 . 2  |-  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  ( U  e.  ( R  ub  A )  ->  ( U  e.  dom  R  /\  A. x  e.  A  x R U ) ) )
5 simprl 732 . . . 4  |-  ( ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  /\  ( U  e.  dom  R  /\  A. x  e.  A  x R U ) )  ->  U  e.  dom  R )
6 ssralv 3237 . . . . . . . 8  |-  ( B 
C_  A  ->  ( A. x  e.  A  x R U  ->  A. x  e.  B  x R U ) )
763ad2ant3 978 . . . . . . 7  |-  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  ( A. x  e.  A  x R U  ->  A. x  e.  B  x R U ) )
87com12 27 . . . . . 6  |-  ( A. x  e.  A  x R U  ->  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  A. x  e.  B  x R U ) )
98adantl 452 . . . . 5  |-  ( ( U  e.  dom  R  /\  A. x  e.  A  x R U )  -> 
( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  A. x  e.  B  x R U ) )
109impcom 419 . . . 4  |-  ( ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  /\  ( U  e.  dom  R  /\  A. x  e.  A  x R U ) )  ->  A. x  e.  B  x R U )
11 simp1 955 . . . . . . 7  |-  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  R  e. PresetRel )
12 ssexg 4160 . . . . . . . . 9  |-  ( ( B  C_  A  /\  A  e.  C )  ->  B  e.  _V )
1312ancoms 439 . . . . . . . 8  |-  ( ( A  e.  C  /\  B  C_  A )  ->  B  e.  _V )
14133adant1 973 . . . . . . 7  |-  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  B  e.  _V )
1511, 14jca 518 . . . . . 6  |-  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  ( R  e. PresetRel  /\  B  e. 
_V ) )
1615adantr 451 . . . . 5  |-  ( ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  /\  ( U  e.  dom  R  /\  A. x  e.  A  x R U ) )  ->  ( R  e. PresetRel  /\  B  e.  _V )
)
171puub2 25258 . . . . 5  |-  ( ( R  e. PresetRel  /\  B  e. 
_V )  ->  ( U  e.  ( R  ub  B )  <->  ( U  e.  dom  R  /\  A. x  e.  B  x R U ) ) )
1816, 17syl 15 . . . 4  |-  ( ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  /\  ( U  e.  dom  R  /\  A. x  e.  A  x R U ) )  ->  ( U  e.  ( R  ub  B
)  <->  ( U  e. 
dom  R  /\  A. x  e.  B  x R U ) ) )
195, 10, 18mpbir2and 888 . . 3  |-  ( ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  /\  ( U  e.  dom  R  /\  A. x  e.  A  x R U ) )  ->  U  e.  ( R  ub  B ) )
2019ex 423 . 2  |-  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  (
( U  e.  dom  R  /\  A. x  e.  A  x R U )  ->  U  e.  ( R  ub  B
) ) )
214, 20syld 40 1  |-  ( ( R  e. PresetRel  /\  A  e.  C  /\  B  C_  A )  ->  ( U  e.  ( R  ub  A )  ->  U  e.  ( R  ub  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152   class class class wbr 4023   dom cdm 4689  (class class class)co 5858  PresetRelcpresetrel 25215    ub cub 25218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-prs 25223  df-ub 25253
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