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Theorem fneer 26368
 Description: Fineness intersected with its converse is an equivalence relation. (Contributed by Jeff Hankins, 6-Oct-2009.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypothesis
Ref Expression
fneval.1
Assertion
Ref Expression
fneer

Proof of Theorem fneer
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5728 . 2
2 fneval.1 . . . . . 6
3 inss1 3561 . . . . . 6
42, 3eqsstri 3378 . . . . 5
5 fnerel 26347 . . . . 5
6 relss 4963 . . . . 5
74, 5, 6mp2 9 . . . 4
8 dfrel4v 5322 . . . 4
97, 8mpbi 200 . . 3
10 vex 2959 . . . . 5
11 vex 2959 . . . . 5
122fneval 26367 . . . . 5
1310, 11, 12mp2an 654 . . . 4
1413opabbii 4272 . . 3
159, 14eqtri 2456 . 2
161, 15eqer 6938 1
 Colors of variables: wff set class Syntax hints:   wb 177   wceq 1652   wcel 1725  cvv 2956   cin 3319   wss 3320   class class class wbr 4212  copab 4265  ccnv 4877   wrel 4883  cfv 5454   wer 6902  ctg 13665  cfne 26339 This theorem is referenced by:  topfneec  26371  topfneec2  26372 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-er 6905  df-topgen 13667  df-fne 26343
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