Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prltub Unicode version

Theorem prltub 25260
Description: If  R is a preset,  U R V and  U is an upper bound of  A then  V is an upper bound of  A. Bourbaki E.III.9 nb 8. (Contributed by FL, 23-May-2011.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
prltub  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  V  e.  ( R  ub  A ) )

Proof of Theorem prltub
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  R  e. PresetRel )
2 simp3r 984 . . 3  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  U R V )
3 eqid 2283 . . . 4  |-  dom  R  =  dom  R
43pre2befi2 25232 . . 3  |-  ( ( R  e. PresetRel  /\  U R V )  ->  V  e.  dom  R )
51, 2, 4syl2anc 642 . 2  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  V  e.  dom  R )
6 simp3l 983 . . . . 5  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  U  e.  ( R  ub  A ) )
73puub2 25258 . . . . . 6  |-  ( ( R  e. PresetRel  /\  A  e.  B )  ->  ( U  e.  ( R  ub  A )  <->  ( U  e.  dom  R  /\  A. b  e.  A  b R U ) ) )
873adant3 975 . . . . 5  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  -> 
( U  e.  ( R  ub  A )  <-> 
( U  e.  dom  R  /\  A. b  e.  A  b R U ) ) )
96, 8mpbid 201 . . . 4  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  -> 
( U  e.  dom  R  /\  A. b  e.  A  b R U ) )
109simprd 449 . . 3  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  A. b  e.  A  b R U )
11 preotr2 25235 . . . . . . 7  |-  ( ( R  e. PresetRel  /\  (
b R U  /\  U R V ) )  ->  b R V )
1211ex 423 . . . . . 6  |-  ( R  e. PresetRel  ->  ( ( b R U  /\  U R V )  ->  b R V ) )
13123ad2ant1 976 . . . . 5  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  -> 
( ( b R U  /\  U R V )  ->  b R V ) )
142, 13mpan2d 655 . . . 4  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  -> 
( b R U  ->  b R V ) )
1514ralimdv 2622 . . 3  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  -> 
( A. b  e.  A  b R U  ->  A. b  e.  A  b R V ) )
1610, 15mpd 14 . 2  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  A. b  e.  A  b R V )
173puub2 25258 . . 3  |-  ( ( R  e. PresetRel  /\  A  e.  B )  ->  ( V  e.  ( R  ub  A )  <->  ( V  e.  dom  R  /\  A. b  e.  A  b R V ) ) )
18173adant3 975 . 2  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  -> 
( V  e.  ( R  ub  A )  <-> 
( V  e.  dom  R  /\  A. b  e.  A  b R V ) ) )
195, 16, 18mpbir2and 888 1  |-  ( ( R  e. PresetRel  /\  A  e.  B  /\  ( U  e.  ( R  ub  A )  /\  U R V ) )  ->  V  e.  ( R  ub  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1684   A.wral 2543   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-reu 2550  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-mpt 4079  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-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-prs 25223  df-ub 25253
  Copyright terms: Public domain W3C validator