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Theorem fndmdif 5945
 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 3535 . . . . 5
2 dmss 4996 . . . . 5
31, 2ax-mp 5 . . . 4
4 fndm 5636 . . . . 5
54adantr 466 . . . 4
63, 5syl5sseq 3455 . . 3
7 dfss1 3610 . . 3
86, 7sylib 199 . 2
9 vex 3025 . . . . 5
109eldm 4994 . . . 4
11 eqcom 2435 . . . . . . . . 9
12 fnbrfvb 5865 . . . . . . . . 9
1311, 12syl5bb 260 . . . . . . . 8
1413adantll 718 . . . . . . 7
1514necon3abid 2637 . . . . . 6
16 fvex 5835 . . . . . . 7
17 breq2 4370 . . . . . . . 8
1817notbid 295 . . . . . . 7
1916, 18ceqsexv 3060 . . . . . 6
2015, 19syl6bbr 266 . . . . 5
21 eqcom 2435 . . . . . . . . . 10
22 fnbrfvb 5865 . . . . . . . . . 10
2321, 22syl5bb 260 . . . . . . . . 9
2423adantlr 719 . . . . . . . 8
2524anbi1d 709 . . . . . . 7
26 brdif 4417 . . . . . . 7
2725, 26syl6bbr 266 . . . . . 6
2827exbidv 1762 . . . . 5
2920, 28bitr2d 257 . . . 4
3010, 29syl5bb 260 . . 3
3130rabbi2dva 3613 . 2
328, 31eqtr3d 2464 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wb 187   wa 370   wceq 1437  wex 1657   wcel 1872   wne 2599  crab 2718   cdif 3376   cin 3378   wss 3379   class class class wbr 4366   cdm 4796   wfn 5539  cfv 5544 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552 This theorem is referenced by:  fndmdifcom  5946  fndmdifeq0  5947  fndifnfp  6052  wemapsolem  8018  wemapso2lem  8020  dsmmbas2  19242  frlmbas  19260  ptcmplem2  21010
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