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Theorem defse3 25272
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 25252 . . 3  |-  ( R  e.  PosetRel  ->  (leR `  R
)  =  ( ge
`  `' R ) )
2 posispre 25241 . . . . . . 7  |-  ( R  e.  PosetRel  ->  R  e. PresetRel )
3 eqid 2283 . . . . . . . . 9  |-  dom  R  =  dom  R
43domcnvpre 25233 . . . . . . . 8  |-  ( R  e. PresetRel  ->  dom  R  =  dom  `' R )
54eqcomd 2288 . . . . . . 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 2300 . . . . . . . . . 10  |-  ( ( dom  `' R  =  dom  R  /\  dom  R  =  X )  ->  dom  `' R  =  X
)
9 cnvps 14321 . . . . . . . . . . . . 13  |-  ( R  e.  PosetRel  ->  `' R  e.  PosetRel )
10 eqid 2283 . . . . . . . . . . . . . . 15  |-  dom  `' R  =  dom  `' R
1110defge3 25271 . . . . . . . . . . . . . 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 2344 . . . . . . . . . . . . 13  |-  ( X  =  dom  `' R  ->  ( ( ge `  `' R )  e.  X  <->  ( ge `  `' R
)  e.  dom  `' R ) )
15 raleq 2736 . . . . . . . . . . . . 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 2286 . . . . . . . . . 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 2286 . . . . . . 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 2791 . . . . . . . 8  |-  x  e. 
_V
25 fvex 5539 . . . . . . . 8  |-  ( ge
`  `' R )  e.  _V
2624, 25brcnv 4864 . . . . . . 7  |-  ( x `' R ( ge `  `' R )  <->  ( ge `  `' R ) R x )
2726bicomi 193 . . . . . 6  |-  ( ( ge `  `' R
) R x  <->  x `' R ( ge `  `' R ) )
2827ralbii 2567 . . . . 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 2343 . . . . 5  |-  ( (leR
`  R )  =  ( ge `  `' R )  ->  (
(leR `  R )  e.  X  <->  ( ge `  `' R )  e.  X
) )
31 breq1 4026 . . . . . 6  |-  ( (leR
`  R )  =  ( ge `  `' R )  ->  (
(leR `  R ) R x  <->  ( ge `  `' R ) R x ) )
3231ralbidv 2563 . . . . 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 1623    e. wcel 1684   A.wral 2543   class class class wbr 4023   `'ccnv 4688   dom cdm 4689   ` cfv 5255   PosetRelcps 14301  PresetRelcpresetrel 25215   gecge 25220  leRcse 25221
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-pow 4188  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-nel 2449  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-undef 6298  df-riota 6304  df-ps 14306  df-spw 14308  df-nfw 14309  df-prs 25223  df-ge 25248  df-ler 25249
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