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Theorem prltub 25363
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 2296 . . . 4  |-  dom  R  =  dom  R
43pre2befi2 25335 . . 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 25361 . . . . . 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 25338 . . . . . . 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 2635 . . 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 25361 . . 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 1696   A.wral 2556   class class class wbr 4039   dom cdm 4705  (class class class)co 5874  PresetRelcpresetrel 25318    ub cub 25321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-prs 25326  df-ub 25356
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