Mathbox for Jeff Hankins < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  filnetlem2 Structured version   Unicode version

Theorem filnetlem2 26409
 Description: Lemma for filnet 26412. The field of the direction. (Contributed by Jeff Hankins, 13-Dec-2009.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypotheses
Ref Expression
filnet.h
filnet.d
Assertion
Ref Expression
filnetlem2
Distinct variable groups:   ,,,   ,,
Allowed substitution hints:   (,,)   ()

Proof of Theorem filnetlem2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 idref 5980 . . 3
2 ssid 3368 . . . . . 6
3 filnet.h . . . . . . 7
4 filnet.d . . . . . . 7
5 vex 2960 . . . . . . 7
63, 4, 5, 5filnetlem1 26408 . . . . . 6
72, 6mpbiran2 887 . . . . 5
87biimpri 199 . . . 4
98anidms 628 . . 3
101, 9mprgbir 2777 . 2
11 opabssxp 4951 . . 3
124, 11eqsstri 3379 . 2
1310, 12pm3.2i 443 1
 Colors of variables: wff set class Syntax hints:   wa 360   wceq 1653   wcel 1726   wss 3321  csn 3815  ciun 4094   class class class wbr 4213  copab 4266   cid 4494   cxp 4877   cres 4881  cfv 5455  c1st 6348 This theorem is referenced by:  filnetlem3  26410  filnetlem4  26411 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463
 Copyright terms: Public domain W3C validator