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Theorem imagesset 30278
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 3460 . . . . . . . 8  |-  y  C_  y
2 sseq2 3463 . . . . . . . . 9  |-  ( z  =  y  ->  (
y  C_  z  <->  y  C_  y ) )
32rspcev 3159 . . . . . . . 8  |-  ( ( y  e.  x  /\  y  C_  y )  ->  E. z  e.  x  y  C_  z )
41, 3mpan2 669 . . . . . . 7  |-  ( y  e.  x  ->  E. z  e.  x  y  C_  z )
5 vex 3061 . . . . . . . . 9  |-  y  e. 
_V
65elima 5161 . . . . . . . 8  |-  ( y  e.  ( `' SSet " x )  <->  E. z  e.  x  z `' SSet y )
7 vex 3061 . . . . . . . . . . 11  |-  z  e. 
_V
87, 5brcnv 5005 . . . . . . . . . 10  |-  ( z `' SSet y  <->  y SSet z )
97brsset 30214 . . . . . . . . . 10  |-  ( y
SSet z  <->  y  C_  z )
108, 9bitri 249 . . . . . . . . 9  |-  ( z `' SSet y  <->  y  C_  z )
1110rexbii 2905 . . . . . . . 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 3445 . . . . 5  |-  x  C_  ( `' SSet " x )
15 sseq2 3463 . . . . 5  |-  ( y  =  ( `' SSet " x )  ->  (
x  C_  y  <->  x  C_  ( `' SSet " x ) ) )
1614, 15mpbiri 233 . . . 4  |-  ( y  =  ( `' SSet " x )  ->  x  C_  y )
17 vex 3061 . . . . . 6  |-  x  e. 
_V
1817, 5brimage 30251 . . . . 5  |-  ( xImage `' SSet y  <->  y  =  ( `' SSet " x ) )
19 df-br 4395 . . . . 5  |-  ( xImage `' SSet y  <->  <. x ,  y >.  e. Image `' SSet )
2018, 19bitr3i 251 . . . 4  |-  ( y  =  ( `' SSet " x )  <->  <. x ,  y >.  e. Image `' SSet )
215brsset 30214 . . . . 5  |-  ( x
SSet y  <->  x  C_  y
)
22 df-br 4395 . . . . 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 1640 . 2  |-  A. x A. y ( <. x ,  y >.  e. Image `' SSet  ->  <. x ,  y
>.  e.  SSet )
26 funimage 30253 . . 3  |-  Fun Image `' SSet
27 funrel 5585 . . 3  |-  ( Fun Image `' SSet  ->  Rel Image `' SSet )
28 ssrel 4911 . . 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 1403    = wceq 1405    e. wcel 1842   E.wrex 2754    C_ wss 3413   <.cop 3977   class class class wbr 4394   `'ccnv 4821   "cima 4825   Rel wrel 4827   Fun wfun 5562   SSetcsset 30156  Imagecimage 30164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-symdif 3669  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-eprel 4733  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-fo 5574  df-fv 5576  df-1st 6783  df-2nd 6784  df-txp 30178  df-sset 30180  df-image 30188
This theorem is referenced by: (None)
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