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Theorem fndmin 5838
 Description: Two ways to express the locus of equality between two functions. (Contributed by Stefan O'Rear, 17-Jan-2015.)
Assertion
Ref Expression
fndmin
Distinct variable groups:   ,   ,   ,

Proof of Theorem fndmin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dffn5 5773 . . . . . . 7
21biimpi 188 . . . . . 6
3 df-mpt 4269 . . . . . 6
42, 3syl6eq 2485 . . . . 5
5 dffn5 5773 . . . . . . 7
65biimpi 188 . . . . . 6
7 df-mpt 4269 . . . . . 6
86, 7syl6eq 2485 . . . . 5
94, 8ineqan12d 3545 . . . 4
10 inopab 5006 . . . 4
119, 10syl6eq 2485 . . 3
1211dmeqd 5073 . 2
13 19.42v 1929 . . . . 5
14 anandi 803 . . . . . 6
1514exbii 1593 . . . . 5
16 fvex 5743 . . . . . . 7
17 eqeq1 2443 . . . . . . 7
1816, 17ceqsexv 2992 . . . . . 6
1918anbi2i 677 . . . . 5
2013, 15, 193bitr3i 268 . . . 4
2120abbii 2549 . . 3
22 dmopab 5081 . . 3
23 df-rab 2715 . . 3
2421, 22, 233eqtr4i 2467 . 2
2512, 24syl6eq 2485 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360  wex 1551   wceq 1653   wcel 1726  cab 2423  crab 2710   cin 3320  copab 4266   cmpt 4267   cdm 4879   wfn 5450  cfv 5455 This theorem is referenced by:  fneqeql  5839  mhmeql  14765  ghmeql  15029  lmhmeql  16132  hauseqlcld  17679  cvmliftmolem1  24969  cvmliftmolem2  24970  fninfp  26736  hausgraph  27509 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-rab 2715  df-v 2959  df-sbc 3163  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-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-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463
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