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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

Proof of Theorem imagesset
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3516 . . . . . . . 8
2 sseq2 3519 . . . . . . . . 9
32rspcev 3207 . . . . . . . 8
41, 3mpan2 671 . . . . . . 7
5 vex 3109 . . . . . . . . 9
65elima 5333 . . . . . . . 8
7 vex 3109 . . . . . . . . . . 11
87, 5brcnv 5176 . . . . . . . . . 10
97brsset 29102 . . . . . . . . . 10
108, 9bitri 249 . . . . . . . . 9
1110rexbii 2958 . . . . . . . 8
126, 11bitri 249 . . . . . . 7
134, 12sylibr 212 . . . . . 6
1413ssriv 3501 . . . . 5
15 sseq2 3519 . . . . 5
1614, 15mpbiri 233 . . . 4
17 vex 3109 . . . . . 6
1817, 5brimage 29139 . . . . 5 Image
19 df-br 4441 . . . . 5 Image Image
2018, 19bitr3i 251 . . . 4 Image
215brsset 29102 . . . . 5
22 df-br 4441 . . . . 5
2321, 22bitr3i 251 . . . 4
2416, 20, 233imtr3i 265 . . 3 Image
2524gen2 1597 . 2 Image
26 funimage 29141 . . 3 Image
27 funrel 5596 . . 3 Image Image
28 ssrel 5082 . . 3 Image Image Image
2926, 27, 28mp2b 10 . 2 Image Image
3025, 29mpbir 209 1 Image
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184  wal 1372   wceq 1374   wcel 1762  wrex 2808   wss 3469  cop 4026   class class class wbr 4440  ccnv 4991  cima 4995   wrel 4997   wfun 5573  csset 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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