Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj1253 Structured version   Unicode version

Theorem bnj1253 34216
Description: Technical lemma for bnj60 34261. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1253.1  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
bnj1253.2  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
bnj1253.3  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
bnj1253.4  |-  D  =  ( dom  g  i^i 
dom  h )
bnj1253.5  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
bnj1253.6  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
bnj1253.7  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
Assertion
Ref Expression
bnj1253  |-  ( ph  ->  E  =/=  (/) )
Distinct variable groups:    A, f    B, f, g    B, h, f    D, d    x, D   
f, G, g    h, G    R, f    g, Y   
h, Y    f, d,
g    h, d    x, f, g    x, h
Allowed substitution hints:    ph( x, y, f, g, h, d)    ps( x, y, f, g, h, d)    A( x, y, g, h, d)    B( x, y, d)    C( x, y, f, g, h, d)    D( y, f, g, h)    R( x, y, g, h, d)    E( x, y, f, g, h, d)    G( x, y, d)    Y( x, y, f, d)

Proof of Theorem bnj1253
StepHypRef Expression
1 bnj1253.6 . . . 4  |-  ( ph  <->  ( R  FrSe  A  /\  g  e.  C  /\  h  e.  C  /\  ( g  |`  D )  =/=  ( h  |`  D ) ) )
21bnj1254 34011 . . 3  |-  ( ph  ->  ( g  |`  D )  =/=  ( h  |`  D ) )
3 bnj1253.1 . . . . . . . . . . 11  |-  B  =  { d  |  ( d  C_  A  /\  A. x  e.  d  pred ( x ,  A ,  R )  C_  d
) }
4 bnj1253.2 . . . . . . . . . . 11  |-  Y  = 
<. x ,  ( f  |`  pred ( x ,  A ,  R ) ) >.
5 bnj1253.3 . . . . . . . . . . 11  |-  C  =  { f  |  E. d  e.  B  (
f  Fn  d  /\  A. x  e.  d  ( f `  x )  =  ( G `  Y ) ) }
6 bnj1253.4 . . . . . . . . . . 11  |-  D  =  ( dom  g  i^i 
dom  h )
7 bnj1253.5 . . . . . . . . . . 11  |-  E  =  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }
8 bnj1253.7 . . . . . . . . . . 11  |-  ( ps  <->  (
ph  /\  x  e.  E  /\  A. y  e.  E  -.  y R x ) )
93, 4, 5, 6, 7, 1, 8bnj1256 34214 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
106bnj1292 34017 . . . . . . . . . . . 12  |-  D  C_  dom  g
11 fndm 5686 . . . . . . . . . . . 12  |-  ( g  Fn  d  ->  dom  g  =  d )
1210, 11syl5sseq 3547 . . . . . . . . . . 11  |-  ( g  Fn  d  ->  D  C_  d )
13 fnssres 5700 . . . . . . . . . . 11  |-  ( ( g  Fn  d  /\  D  C_  d )  -> 
( g  |`  D )  Fn  D )
1412, 13mpdan 668 . . . . . . . . . 10  |-  ( g  Fn  d  ->  (
g  |`  D )  Fn  D )
159, 14bnj31 33915 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( g  |`  D )  Fn  D )
1615bnj1265 34014 . . . . . . . 8  |-  ( ph  ->  ( g  |`  D )  Fn  D )
173, 4, 5, 6, 7, 1, 8bnj1259 34215 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
186bnj1293 34018 . . . . . . . . . . . 12  |-  D  C_  dom  h
19 fndm 5686 . . . . . . . . . . . 12  |-  ( h  Fn  d  ->  dom  h  =  d )
2018, 19syl5sseq 3547 . . . . . . . . . . 11  |-  ( h  Fn  d  ->  D  C_  d )
21 fnssres 5700 . . . . . . . . . . 11  |-  ( ( h  Fn  d  /\  D  C_  d )  -> 
( h  |`  D )  Fn  D )
2220, 21mpdan 668 . . . . . . . . . 10  |-  ( h  Fn  d  ->  (
h  |`  D )  Fn  D )
2317, 22bnj31 33915 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( h  |`  D )  Fn  D )
2423bnj1265 34014 . . . . . . . 8  |-  ( ph  ->  ( h  |`  D )  Fn  D )
25 ssid 3518 . . . . . . . . 9  |-  D  C_  D
26 fvreseq 5990 . . . . . . . . 9  |-  ( ( ( ( g  |`  D )  Fn  D  /\  ( h  |`  D )  Fn  D )  /\  D  C_  D )  -> 
( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
2725, 26mpan2 671 . . . . . . . 8  |-  ( ( ( g  |`  D )  Fn  D  /\  (
h  |`  D )  Fn  D )  ->  (
( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
2816, 24, 27syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
29 residm 5315 . . . . . . . 8  |-  ( ( g  |`  D )  |`  D )  =  ( g  |`  D )
30 residm 5315 . . . . . . . 8  |-  ( ( h  |`  D )  |`  D )  =  ( h  |`  D )
3129, 30eqeq12i 2477 . . . . . . 7  |-  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
32 df-ral 2812 . . . . . . 7  |-  ( A. x  e.  D  (
( g  |`  D ) `
 x )  =  ( ( h  |`  D ) `  x
)  <->  A. x ( x  e.  D  ->  (
( g  |`  D ) `
 x )  =  ( ( h  |`  D ) `  x
) ) )
3328, 31, 323bitr3g 287 . . . . . 6  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) ) ) )
34 fvres 5886 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( g  |`  D ) `
 x )  =  ( g `  x
) )
35 fvres 5886 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( h  |`  D ) `
 x )  =  ( h `  x
) )
3634, 35eqeq12d 2479 . . . . . . . 8  |-  ( x  e.  D  ->  (
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x )  <->  ( g `  x )  =  ( h `  x ) ) )
3736pm5.74i 245 . . . . . . 7  |-  ( ( x  e.  D  -> 
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) )  <->  ( x  e.  D  ->  ( g `
 x )  =  ( h `  x
) ) )
3837albii 1641 . . . . . 6  |-  ( A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) )
3933, 38syl6bb 261 . . . . 5  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) ) )
4039necon3abid 2703 . . . 4  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) ) )
41 df-rex 2813 . . . . 5  |-  ( E. x  e.  D  ( g `  x )  =/=  ( h `  x )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
42 pm4.61 426 . . . . . . 7  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
43 df-ne 2654 . . . . . . . 8  |-  ( ( g `  x )  =/=  ( h `  x )  <->  -.  (
g `  x )  =  ( h `  x ) )
4443anbi2i 694 . . . . . . 7  |-  ( ( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
4542, 44bitr4i 252 . . . . . 6  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
4645exbii 1668 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
47 exnal 1649 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) )
4841, 46, 473bitr2ri 274 . . . 4  |-  ( -. 
A. x ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
)
4940, 48syl6bb 261 . . 3  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
) )
502, 49mpbid 210 . 2  |-  ( ph  ->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
517neeq1i 2742 . . 3  |-  ( E  =/=  (/)  <->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
52 rabn0 3814 . . 3  |-  ( { x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5351, 52bitri 249 . 2  |-  ( E  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5450, 53sylibr 212 1  |-  ( ph  ->  E  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973   A.wal 1393    = wceq 1395   E.wex 1613    e. wcel 1819   {cab 2442    =/= wne 2652   A.wral 2807   E.wrex 2808   {crab 2811    i^i cin 3470    C_ wss 3471   (/)c0 3793   <.cop 4038   class class class wbr 4456   dom cdm 5008    |` cres 5010    Fn wfn 5589   ` cfv 5594    /\ w-bnj17 33881    predc-bnj14 33883    FrSe w-bnj15 33887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-bnj17 33882
This theorem is referenced by:  bnj1311  34223
  Copyright terms: Public domain W3C validator