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Theorem defse3 25375
Description: The least element of a poset is an element, when it exists, that is less than the other elements of the poset. Use the idiom  (leR `  R
)  e.  X when you mean the least element of  X exists. (Contributed by FL, 30-Dec-2011.)
Hypothesis
Ref Expression
defse3.1  |-  X  =  dom  R
Assertion
Ref Expression
defse3  |-  ( ( R  e.  PosetRel  /\  (leR `  R )  e.  X
)  ->  A. x  e.  X  (leR `  R
) R x )
Distinct variable groups:    x, R    x, X

Proof of Theorem defse3
StepHypRef Expression
1 sege 25355 . . 3  |-  ( R  e.  PosetRel  ->  (leR `  R
)  =  ( ge
`  `' R ) )
2 posispre 25344 . . . . . . 7  |-  ( R  e.  PosetRel  ->  R  e. PresetRel )
3 eqid 2296 . . . . . . . . 9  |-  dom  R  =  dom  R
43domcnvpre 25336 . . . . . . . 8  |-  ( R  e. PresetRel  ->  dom  R  =  dom  `' R )
54eqcomd 2301 . . . . . . 7  |-  ( R  e. PresetRel  ->  dom  `' R  =  dom  R )
62, 5syl 15 . . . . . 6  |-  ( R  e.  PosetRel  ->  dom  `' R  =  dom  R )
7 defse3.1 . . . . . . 7  |-  X  =  dom  R
8 eqtr 2313 . . . . . . . . . 10  |-  ( ( dom  `' R  =  dom  R  /\  dom  R  =  X )  ->  dom  `' R  =  X
)
9 cnvps 14337 . . . . . . . . . . . . 13  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
10 eqid 2296 . . . . . . . . . . . . . . 15  |-  dom  `' R  =  dom  `' R
1110defge3 25374 . . . . . . . . . . . . . 14  |-  ( ( `' R  e.  PosetRel  /\  ( ge `  `' R )  e.  dom  `' R
)  ->  A. x  e.  dom  `' R x `' R ( ge `  `' R ) )
1211ex 423 . . . . . . . . . . . . 13  |-  ( `' R  e.  PosetRel  ->  (
( ge `  `' R )  e.  dom  `' R  ->  A. x  e.  dom  `' R x `' R ( ge `  `' R ) ) )
139, 12syl 15 . . . . . . . . . . . 12  |-  ( R  e.  PosetRel  ->  ( ( ge
`  `' R )  e.  dom  `' R  ->  A. x  e.  dom  `' R x `' R
( ge `  `' R ) ) )
14 eleq2 2357 . . . . . . . . . . . . 13  |-  ( X  =  dom  `' R  ->  ( ( ge `  `' R )  e.  X  <->  ( ge `  `' R
)  e.  dom  `' R ) )
15 raleq 2749 . . . . . . . . . . . . 13  |-  ( X  =  dom  `' R  ->  ( A. x  e.  X  x `' R
( ge `  `' R )  <->  A. x  e.  dom  `' R x `' R ( ge `  `' R ) ) )
1614, 15imbi12d 311 . . . . . . . . . . . 12  |-  ( X  =  dom  `' R  ->  ( ( ( ge
`  `' R )  e.  X  ->  A. x  e.  X  x `' R ( ge `  `' R ) )  <->  ( ( ge `  `' R )  e.  dom  `' R  ->  A. x  e.  dom  `' R x `' R
( ge `  `' R ) ) ) )
1713, 16syl5ibr 212 . . . . . . . . . . 11  |-  ( X  =  dom  `' R  ->  ( R  e.  PosetRel  -> 
( ( ge `  `' R )  e.  X  ->  A. x  e.  X  x `' R ( ge `  `' R ) ) ) )
1817eqcoms 2299 . . . . . . . . . 10  |-  ( dom  `' R  =  X  ->  ( R  e.  PosetRel  -> 
( ( ge `  `' R )  e.  X  ->  A. x  e.  X  x `' R ( ge `  `' R ) ) ) )
198, 18syl 15 . . . . . . . . 9  |-  ( ( dom  `' R  =  dom  R  /\  dom  R  =  X )  -> 
( R  e.  PosetRel  -> 
( ( ge `  `' R )  e.  X  ->  A. x  e.  X  x `' R ( ge `  `' R ) ) ) )
2019expcom 424 . . . . . . . 8  |-  ( dom 
R  =  X  -> 
( dom  `' R  =  dom  R  ->  ( R  e.  PosetRel  ->  ( ( ge `  `' R
)  e.  X  ->  A. x  e.  X  x `' R ( ge `  `' R ) ) ) ) )
2120eqcoms 2299 . . . . . . 7  |-  ( X  =  dom  R  -> 
( dom  `' R  =  dom  R  ->  ( R  e.  PosetRel  ->  ( ( ge `  `' R
)  e.  X  ->  A. x  e.  X  x `' R ( ge `  `' R ) ) ) ) )
227, 21ax-mp 8 . . . . . 6  |-  ( dom  `' R  =  dom  R  ->  ( R  e.  PosetRel 
->  ( ( ge `  `' R )  e.  X  ->  A. x  e.  X  x `' R ( ge `  `' R ) ) ) )
236, 22mpcom 32 . . . . 5  |-  ( R  e.  PosetRel  ->  ( ( ge
`  `' R )  e.  X  ->  A. x  e.  X  x `' R ( ge `  `' R ) ) )
24 vex 2804 . . . . . . . 8  |-  x  e. 
_V
25 fvex 5555 . . . . . . . 8  |-  ( ge
`  `' R )  e.  _V
2624, 25brcnv 4880 . . . . . . 7  |-  ( x `' R ( ge `  `' R )  <->  ( ge `  `' R ) R x )
2726bicomi 193 . . . . . 6  |-  ( ( ge `  `' R
) R x  <->  x `' R ( ge `  `' R ) )
2827ralbii 2580 . . . . 5  |-  ( A. x  e.  X  ( ge `  `' R ) R x  <->  A. x  e.  X  x `' R ( ge `  `' R ) )
2923, 28syl6ibr 218 . . . 4  |-  ( R  e.  PosetRel  ->  ( ( ge
`  `' R )  e.  X  ->  A. x  e.  X  ( ge `  `' R ) R x ) )
30 eleq1 2356 . . . . 5  |-  ( (leR
`  R )  =  ( ge `  `' R )  ->  (
(leR `  R )  e.  X  <->  ( ge `  `' R )  e.  X
) )
31 breq1 4042 . . . . . 6  |-  ( (leR
`  R )  =  ( ge `  `' R )  ->  (
(leR `  R ) R x  <->  ( ge `  `' R ) R x ) )
3231ralbidv 2576 . . . . 5  |-  ( (leR
`  R )  =  ( ge `  `' R )  ->  ( A. x  e.  X  (leR `  R ) R x  <->  A. x  e.  X  ( ge `  `' R
) R x ) )
3330, 32imbi12d 311 . . . 4  |-  ( (leR
`  R )  =  ( ge `  `' R )  ->  (
( (leR `  R
)  e.  X  ->  A. x  e.  X  (leR `  R ) R x )  <->  ( ( ge `  `' R )  e.  X  ->  A. x  e.  X  ( ge `  `' R ) R x ) ) )
3429, 33syl5ibr 212 . . 3  |-  ( (leR
`  R )  =  ( ge `  `' R )  ->  ( R  e.  PosetRel  ->  ( (leR
`  R )  e.  X  ->  A. x  e.  X  (leR `  R
) R x ) ) )
351, 34mpcom 32 . 2  |-  ( R  e.  PosetRel  ->  ( (leR `  R )  e.  X  ->  A. x  e.  X  (leR `  R ) R x ) )
3635imp 418 1  |-  ( ( R  e.  PosetRel  /\  (leR `  R )  e.  X
)  ->  A. x  e.  X  (leR `  R
) R x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ` cfv 5271   PosetRelcps 14317  PresetRelcpresetrel 25318   gecge 25323  leRcse 25324
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-pow 4204  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-nel 2462  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-undef 6314  df-riota 6320  df-ps 14322  df-spw 14324  df-nfw 14325  df-prs 25326  df-ge 25351  df-ler 25352
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