Theorem imagesset 32060

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.

Step	Hyp	Ref	Expression
1		ssid 3624	. . . . . . . 8 |- y C_ y
2		sseq2 3627	. . . . . . . . 9 |- ( z = y -> ( y C_ z <-> y C_ y ) )
3	2	rspcev 3309	. . . . . . . 8 |- ( ( y e. x /\ y C_ y ) -> E. z e. x y C_ z )
4	1, 3	mpan2 707	. . . . . . 7 |- ( y e. x -> E. z e. x y C_ z )
5		vex 3203	. . . . . . . . 9 |- y e. _V
6	5	elima 5471	. . . . . . . 8 |- ( y e. ( `' SSet " x ) <-> E. z e. x z `' SSet y )
7		vex 3203	. . . . . . . . . . 11 |- z e. _V
8	7, 5	brcnv 5305	. . . . . . . . . 10 |- ( z `' SSet y <-> y SSet z )
9	7	brsset 31996	. . . . . . . . . 10 |- ( y SSet z <-> y C_ z )
10	8, 9	bitri 264	. . . . . . . . 9 |- ( z `' SSet y <-> y C_ z )
11	10	rexbii 3041	. . . . . . . 8 |- ( E. z e. x z `' SSet y <-> E. z e. x y C_ z )
12	6, 11	bitri 264	. . . . . . 7 |- ( y e. ( `' SSet " x ) <-> E. z e. x y C_ z )
13	4, 12	sylibr 224	. . . . . 6 |- ( y e. x -> y e. ( `' SSet " x ) )
14	13	ssriv 3607	. . . . 5 |- x C_ ( `' SSet " x )
15		sseq2 3627	. . . . 5 |- ( y = ( `' SSet " x ) -> ( x C_ y <-> x C_ ( `' SSet " x ) ) )
16	14, 15	mpbiri 248	. . . 4 |- ( y = ( `' SSet " x ) -> x C_ y )
17		vex 3203	. . . . . 6 |- x e. _V
18	17, 5	brimage 32033	. . . . 5 |- ( xImage `' SSet y <-> y = ( `' SSet " x ) )
19		df-br 4654	. . . . 5 |- ( xImage `' SSet y <-> <. x , y >. e. Image `' SSet )
20	18, 19	bitr3i 266	. . . 4 |- ( y = ( `' SSet " x ) <-> <. x , y >. e. Image `' SSet )
21	5	brsset 31996	. . . . 5 |- ( x SSet y <-> x C_ y )
22		df-br 4654	. . . . 5 |- ( x SSet y <-> <. x , y >. e. SSet )
23	21, 22	bitr3i 266	. . . 4 |- ( x C_ y <-> <. x , y >. e. SSet )
24	16, 20, 23	3imtr3i 280	. . 3 |- ( <. x , y >. e. Image `' SSet -> <. x , y >. e. SSet )
25	24	gen2 1723	. 2 |- A. x A. y ( <. x , y >. e. Image `' SSet -> <. x , y >. e. SSet )
26		funimage 32035	. . 3 |- Fun Image `' SSet
27		funrel 5905	. . 3 |- ( Fun Image `' SSet -> Rel Image `' SSet )
28		ssrel 5207	. . 3 |- ( Rel Image `' SSet -> (Image `' SSet C_ SSet <-> A. x A. y ( <. x , y >. e. Image `' SSet -> <. x , y >. e. SSet ) ) )
29	26, 27, 28	mp2b 10	. 2 |- (Image `' SSet C_ SSet <-> A. x A. y ( <. x , y >. e. Image `' SSet -> <. x , y >. e. SSet ) )
30	25, 29	mpbir 221	1 |- Image `' SSet C_ SSet

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   <.cop 4183   class class class wbr 4653   `'ccnv 5113   "cima 5117   Rel wrel 5119   Fun wfun 5882   SSetcsset 31939  Imagecimage 31947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-symdif 3844  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-eprel 5029  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-sset 31963  df-image 31971

This theorem is referenced by: (None)