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Theorem fndmdif 5863
 Description: Two ways to express the locus of differences between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmdif
Distinct variable groups:   ,   ,   ,

Proof of Theorem fndmdif
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 difss 3460 . . . . 5
2 dmss 5098 . . . . 5
31, 2ax-mp 5 . . . 4
4 fndm 5573 . . . . 5
54adantr 453 . . . 4
63, 5syl5sseq 3382 . . 3
7 dfss1 3531 . . 3
86, 7sylib 190 . 2
9 vex 2965 . . . . 5
109eldm 5096 . . . 4
11 eqcom 2444 . . . . . . . . 9
12 fnbrfvb 5796 . . . . . . . . 9
1311, 12syl5bb 250 . . . . . . . 8
1413adantll 696 . . . . . . 7
1514necon3abid 2640 . . . . . 6
16 fvex 5771 . . . . . . 7
17 breq2 4241 . . . . . . . 8
1817notbid 287 . . . . . . 7
1916, 18ceqsexv 2997 . . . . . 6
2015, 19syl6bbr 256 . . . . 5
21 eqcom 2444 . . . . . . . . . 10
22 fnbrfvb 5796 . . . . . . . . . 10
2321, 22syl5bb 250 . . . . . . . . 9
2423adantlr 697 . . . . . . . 8
2524anbi1d 687 . . . . . . 7
26 brdif 4285 . . . . . . 7
2725, 26syl6bbr 256 . . . . . 6
2827exbidv 1637 . . . . 5
2920, 28bitr2d 247 . . . 4
3010, 29syl5bb 250 . . 3
3130rabbi2dva 3534 . 2
328, 31eqtr3d 2476 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 178   wa 360  wex 1551   wceq 1653   wcel 1727   wne 2605  crab 2715   cdif 3303   cin 3305   wss 3306   class class class wbr 4237   cdm 4907   wfn 5478  cfv 5483 This theorem is referenced by:  fndmdifcom  5864  fndmdifeq0  5865  wemapso2lem  7548  wemapso2  7550  ptcmplem2  18115  fndifnfp  26775  dsmmbas2  27218  frlmbas  27238 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432 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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fn 5486  df-fv 5491
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