Mathbox for Frédéric Liné < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  issubcat Unicode version

Theorem issubcat 25948
 Description: The set of all the subcategories of . (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
issubcat.2
issubcat.3
issubcat.4
issubcat.5
Assertion
Ref Expression
issubcat

Proof of Theorem issubcat
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . . . . . 7
2 issubcat.2 . . . . . . 7
31, 2syl6eqr 2346 . . . . . 6
43pweqd 3643 . . . . 5
5 fveq2 5541 . . . . . . 7
6 issubcat.3 . . . . . . 7
75, 6syl6eqr 2346 . . . . . 6
87pweqd 3643 . . . . 5
94, 8xpeq12d 4730 . . . 4
10 fveq2 5541 . . . . . . 7
11 issubcat.5 . . . . . . 7
1210, 11syl6eqr 2346 . . . . . 6
1312pweqd 3643 . . . . 5
14 fveq2 5541 . . . . . . 7
15 issubcat.4 . . . . . . 7
1614, 15syl6eqr 2346 . . . . . 6
1716pweqd 3643 . . . . 5
1813, 17xpeq12d 4730 . . . 4
199, 18xpeq12d 4730 . . 3
2019ineq2d 3383 . 2
21 df-subcat 25947 . 2
22 fvex 5555 . . . . . . 7
232, 22eqeltri 2366 . . . . . 6
2423pwex 4209 . . . . 5
25 fvex 5555 . . . . . . 7
266, 25eqeltri 2366 . . . . . 6
2726pwex 4209 . . . . 5
2824, 27xpex 4817 . . . 4
29 fvex 5555 . . . . . . 7
3011, 29eqeltri 2366 . . . . . 6
3130pwex 4209 . . . . 5
32 fvex 5555 . . . . . . 7
3315, 32eqeltri 2366 . . . . . 6
3433pwex 4209 . . . . 5
3531, 34xpex 4817 . . . 4
3628, 35xpex 4817 . . 3
3736inex2 4172 . 2
3820, 21, 37fvmpt 5618 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1632   wcel 1696  cvv 2801   cin 3164  cpw 3638   cxp 4703  cfv 5271  cdom_ 25815  ccod_ 25816  cid_ 25817  co_ 25818   ccatOLD 25855   csubcat 25946 This theorem is referenced by:  issubcata  25949 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-subcat 25947
 Copyright terms: Public domain W3C validator