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Theorem imagesset 29166
Description: The Image functor applied to the converse of the subset relationship yields a subset of the subset relationship. (Contributed by Scott Fenton, 14-Apr-2018.)
Assertion
Ref Expression
imagesset  |- Image `' SSet  C_ 
SSet

Proof of Theorem imagesset
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3516 . . . . . . . 8  |-  y  C_  y
2 sseq2 3519 . . . . . . . . 9  |-  ( z  =  y  ->  (
y  C_  z  <->  y  C_  y ) )
32rspcev 3207 . . . . . . . 8  |-  ( ( y  e.  x  /\  y  C_  y )  ->  E. z  e.  x  y  C_  z )
41, 3mpan2 671 . . . . . . 7  |-  ( y  e.  x  ->  E. z  e.  x  y  C_  z )
5 vex 3109 . . . . . . . . 9  |-  y  e. 
_V
65elima 5333 . . . . . . . 8  |-  ( y  e.  ( `' SSet " x )  <->  E. z  e.  x  z `' SSet y )
7 vex 3109 . . . . . . . . . . 11  |-  z  e. 
_V
87, 5brcnv 5176 . . . . . . . . . 10  |-  ( z `' SSet y  <->  y SSet z )
97brsset 29102 . . . . . . . . . 10  |-  ( y
SSet z  <->  y  C_  z )
108, 9bitri 249 . . . . . . . . 9  |-  ( z `' SSet y  <->  y  C_  z )
1110rexbii 2958 . . . . . . . 8  |-  ( E. z  e.  x  z `' SSet y  <->  E. z  e.  x  y  C_  z )
126, 11bitri 249 . . . . . . 7  |-  ( y  e.  ( `' SSet " x )  <->  E. z  e.  x  y  C_  z )
134, 12sylibr 212 . . . . . 6  |-  ( y  e.  x  ->  y  e.  ( `' SSet " x
) )
1413ssriv 3501 . . . . 5  |-  x  C_  ( `' SSet " x )
15 sseq2 3519 . . . . 5  |-  ( y  =  ( `' SSet " x )  ->  (
x  C_  y  <->  x  C_  ( `' SSet " x ) ) )
1614, 15mpbiri 233 . . . 4  |-  ( y  =  ( `' SSet " x )  ->  x  C_  y )
17 vex 3109 . . . . . 6  |-  x  e. 
_V
1817, 5brimage 29139 . . . . 5  |-  ( xImage `' SSet y  <->  y  =  ( `' SSet " x ) )
19 df-br 4441 . . . . 5  |-  ( xImage `' SSet y  <->  <. x ,  y >.  e. Image `' SSet )
2018, 19bitr3i 251 . . . 4  |-  ( y  =  ( `' SSet " x )  <->  <. x ,  y >.  e. Image `' SSet )
215brsset 29102 . . . . 5  |-  ( x
SSet y  <->  x  C_  y
)
22 df-br 4441 . . . . 5  |-  ( x
SSet y  <->  <. x ,  y >.  e.  SSet )
2321, 22bitr3i 251 . . . 4  |-  ( x 
C_  y  <->  <. x ,  y >.  e.  SSet )
2416, 20, 233imtr3i 265 . . 3  |-  ( <.
x ,  y >.  e. Image `' SSet  ->  <. x ,  y >.  e.  SSet )
2524gen2 1597 . 2  |-  A. x A. y ( <. x ,  y >.  e. Image `' SSet  ->  <. x ,  y
>.  e.  SSet )
26 funimage 29141 . . 3  |-  Fun Image `' SSet
27 funrel 5596 . . 3  |-  ( Fun Image `' SSet  ->  Rel Image `' SSet )
28 ssrel 5082 . . 3  |-  ( Rel Image `' SSet  ->  (Image `' SSet  C_  SSet  <->  A. x A. y
( <. x ,  y
>.  e. Image `' SSet  ->  <. x ,  y >.  e.  SSet ) ) )
2926, 27, 28mp2b 10 . 2  |-  (Image `' SSet  C_  SSet  <->  A. x A. y
( <. x ,  y
>.  e. Image `' SSet  ->  <. x ,  y >.  e.  SSet ) )
3025, 29mpbir 209 1  |- Image `' SSet  C_ 
SSet
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   A.wal 1372    = wceq 1374    e. wcel 1762   E.wrex 2808    C_ wss 3469   <.cop 4026   class class class wbr 4440   `'ccnv 4991   "cima 4995   Rel wrel 4997   Fun wfun 5573   SSetcsset 29044  Imagecimage 29052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-eprel 4784  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fo 5585  df-fv 5587  df-1st 6774  df-2nd 6775  df-symdif 29031  df-txp 29066  df-sset 29068  df-image 29076
This theorem is referenced by: (None)
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