Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiin2g Unicode version

Theorem dfiin2g 4363
 Description: Alternate definition of indexed intersection when is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)
Assertion
Ref Expression
dfiin2g
Distinct variable groups:   ,   ,   ,

Proof of Theorem dfiin2g
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ral 2812 . . . 4
2 df-ral 2812 . . . . . 6
3 eleq2 2530 . . . . . . . . . . . . 13
43biimprcd 225 . . . . . . . . . . . 12
54alrimiv 1719 . . . . . . . . . . 11
6 eqid 2457 . . . . . . . . . . . 12
7 eqeq1 2461 . . . . . . . . . . . . . 14
87, 3imbi12d 320 . . . . . . . . . . . . 13
98spcgv 3194 . . . . . . . . . . . 12
106, 9mpii 43 . . . . . . . . . . 11
115, 10impbid2 204 . . . . . . . . . 10
1211imim2i 14 . . . . . . . . 9
1312pm5.74d 247 . . . . . . . 8
1413alimi 1633 . . . . . . 7
15 albi 1639 . . . . . . 7
1614, 15syl 16 . . . . . 6
172, 16sylbi 195 . . . . 5
18 df-ral 2812 . . . . . . . 8
1918albii 1640 . . . . . . 7
20 alcom 1845 . . . . . . 7
2119, 20bitr4i 252 . . . . . 6
22 r19.23v 2937 . . . . . . . 8
23 vex 3112 . . . . . . . . . 10
24 eqeq1 2461 . . . . . . . . . . 11
2524rexbidv 2968 . . . . . . . . . 10
2623, 25elab 3246 . . . . . . . . 9
2726imbi1i 325 . . . . . . . 8
2822, 27bitr4i 252 . . . . . . 7
2928albii 1640 . . . . . 6
30 19.21v 1729 . . . . . . 7
3130albii 1640 . . . . . 6
3221, 29, 313bitr3ri 276 . . . . 5
3317, 32syl6bb 261 . . . 4
341, 33syl5bb 257 . . 3
3534abbidv 2593 . 2
36 df-iin 4333 . 2
37 df-int 4287 . 2
3835, 36, 373eqtr4g 2523 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808  |^|cint 4286  |^|_ciin 4331 This theorem is referenced by:  dfiin2  4365  iinexg  4612  dfiin3g  5261  iinfi  7897  mreiincl  14993  iinopn  19411  clsval2  19551  alexsublem  20544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-int 4287  df-iin 4333
 Copyright terms: Public domain W3C validator