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Theorem bnj1253 29323
Description: Technical lemma for bnj60 29368. 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 29118 . . 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 29321 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  g  Fn  d )
106bnj1292 29124 . . . . . . . . . . . 12  |-  D  C_  dom  g
11 fndm 5536 . . . . . . . . . . . 12  |-  ( g  Fn  d  ->  dom  g  =  d )
1210, 11syl5sseq 3388 . . . . . . . . . . 11  |-  ( g  Fn  d  ->  D  C_  d )
13 fnssres 5550 . . . . . . . . . . 11  |-  ( ( g  Fn  d  /\  D  C_  d )  -> 
( g  |`  D )  Fn  D )
1412, 13mpdan 650 . . . . . . . . . 10  |-  ( g  Fn  d  ->  (
g  |`  D )  Fn  D )
159, 14bnj31 29021 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( g  |`  D )  Fn  D )
1615bnj1265 29121 . . . . . . . 8  |-  ( ph  ->  ( g  |`  D )  Fn  D )
173, 4, 5, 6, 7, 1, 8bnj1259 29322 . . . . . . . . . 10  |-  ( ph  ->  E. d  e.  B  h  Fn  d )
186bnj1293 29125 . . . . . . . . . . . 12  |-  D  C_  dom  h
19 fndm 5536 . . . . . . . . . . . 12  |-  ( h  Fn  d  ->  dom  h  =  d )
2018, 19syl5sseq 3388 . . . . . . . . . . 11  |-  ( h  Fn  d  ->  D  C_  d )
21 fnssres 5550 . . . . . . . . . . 11  |-  ( ( h  Fn  d  /\  D  C_  d )  -> 
( h  |`  D )  Fn  D )
2220, 21mpdan 650 . . . . . . . . . 10  |-  ( h  Fn  d  ->  (
h  |`  D )  Fn  D )
2317, 22bnj31 29021 . . . . . . . . 9  |-  ( ph  ->  E. d  e.  B  ( h  |`  D )  Fn  D )
2423bnj1265 29121 . . . . . . . 8  |-  ( ph  ->  ( h  |`  D )  Fn  D )
25 ssid 3359 . . . . . . . . 9  |-  D  C_  D
26 fvreseq 5825 . . . . . . . . 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 653 . . . . . . . 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 643 . . . . . . 7  |-  ( ph  ->  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  A. x  e.  D  ( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) ) )
29 residm 5169 . . . . . . . 8  |-  ( ( g  |`  D )  |`  D )  =  ( g  |`  D )
30 residm 5169 . . . . . . . 8  |-  ( ( h  |`  D )  |`  D )  =  ( h  |`  D )
3129, 30eqeq12i 2448 . . . . . . 7  |-  ( ( ( g  |`  D )  |`  D )  =  ( ( h  |`  D )  |`  D )  <->  ( g  |`  D )  =  ( h  |`  D )
)
32 df-ral 2702 . . . . . . 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 279 . . . . . 6  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) ) ) )
34 fvres 5737 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( g  |`  D ) `
 x )  =  ( g `  x
) )
35 fvres 5737 . . . . . . . . 9  |-  ( x  e.  D  ->  (
( h  |`  D ) `
 x )  =  ( h `  x
) )
3634, 35eqeq12d 2449 . . . . . . . 8  |-  ( x  e.  D  ->  (
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x )  <->  ( g `  x )  =  ( h `  x ) ) )
3736pm5.74i 237 . . . . . . 7  |-  ( ( x  e.  D  -> 
( ( g  |`  D ) `  x
)  =  ( ( h  |`  D ) `  x ) )  <->  ( x  e.  D  ->  ( g `
 x )  =  ( h `  x
) ) )
3837albii 1575 . . . . . 6  |-  ( A. x ( x  e.  D  ->  ( (
g  |`  D ) `  x )  =  ( ( h  |`  D ) `
 x ) )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) )
3933, 38syl6bb 253 . . . . 5  |-  ( ph  ->  ( ( g  |`  D )  =  ( h  |`  D )  <->  A. x ( x  e.  D  ->  ( g `  x )  =  ( h `  x ) ) ) )
4039necon3abid 2631 . . . 4  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) ) )
41 df-rex 2703 . . . . 5  |-  ( E. x  e.  D  ( g `  x )  =/=  ( h `  x )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
42 pm4.61 416 . . . . . . 7  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
43 df-ne 2600 . . . . . . . 8  |-  ( ( g `  x )  =/=  ( h `  x )  <->  -.  (
g `  x )  =  ( h `  x ) )
4443anbi2i 676 . . . . . . 7  |-  ( ( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) )  <-> 
( x  e.  D  /\  -.  ( g `  x )  =  ( h `  x ) ) )
4542, 44bitr4i 244 . . . . . 6  |-  ( -.  ( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) )  <-> 
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
4645exbii 1592 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x
( x  e.  D  /\  ( g `  x
)  =/=  ( h `
 x ) ) )
47 exnal 1583 . . . . 5  |-  ( E. x  -.  ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  -.  A. x
( x  e.  D  ->  ( g `  x
)  =  ( h `
 x ) ) )
4841, 46, 473bitr2ri 266 . . . 4  |-  ( -. 
A. x ( x  e.  D  ->  (
g `  x )  =  ( h `  x ) )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
)
4940, 48syl6bb 253 . . 3  |-  ( ph  ->  ( ( g  |`  D )  =/=  (
h  |`  D )  <->  E. x  e.  D  ( g `  x )  =/=  (
h `  x )
) )
502, 49mpbid 202 . 2  |-  ( ph  ->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
517neeq1i 2608 . . 3  |-  ( E  =/=  (/)  <->  { x  e.  D  |  ( g `  x )  =/=  (
h `  x ) }  =/=  (/) )
52 rabn0 3639 . . 3  |-  ( { x  e.  D  | 
( g `  x
)  =/=  ( h `
 x ) }  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5351, 52bitri 241 . 2  |-  ( E  =/=  (/)  <->  E. x  e.  D  ( g `  x
)  =/=  ( h `
 x ) )
5450, 53sylibr 204 1  |-  ( ph  ->  E  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421    =/= wne 2598   A.wral 2697   E.wrex 2698   {crab 2701    i^i cin 3311    C_ wss 3312   (/)c0 3620   <.cop 3809   class class class wbr 4204   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446    /\ w-bnj17 28987    predc-bnj14 28989    FrSe w-bnj15 28993
This theorem is referenced by:  bnj1311  29330
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-bnj17 28988
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