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Theorem reldv 19757
 Description: The derivative function is a relation. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
reldv

Proof of Theorem reldv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4983 . . . . . . . 8 lim
21rgenw 2773 . . . . . . 7 fldt lim
3 reliun 4995 . . . . . . 7 fldt lim fldt lim
42, 3mpbir 201 . . . . . 6 fldt lim
5 df-rel 4885 . . . . . 6 fldt lim fldt lim
64, 5mpbi 200 . . . . 5 fldt lim
76rgenw 2773 . . . 4 fldt lim
87rgenw 2773 . . 3 fldt lim
9 df-dv 19754 . . . 4 fldt lim
109ovmptss 6428 . . 3 fldt lim
118, 10ax-mp 8 . 2
12 df-rel 4885 . 2
1311, 12mpbir 201 1
 Colors of variables: wff set class Syntax hints:  wral 2705  cvv 2956   cdif 3317   wss 3320  cpw 3799  csn 3814  ciun 4093   cmpt 4266   cxp 4876   cdm 4878   wrel 4883  cfv 5454  (class class class)co 6081   cpm 7019  cc 8988   cmin 9291   cdiv 9677   ↾t crest 13648  ctopn 13649  ℂfldccnfld 16703  cnt 17081   lim climc 19749   cdv 19750 This theorem is referenced by:  perfdvf  19790  dvres  19798  dvres3  19800  dvres3a  19801  dvidlem  19802  dvmulbr  19825  dvaddf  19828  dvmulf  19829  dvcobr  19832  dvcof  19834  dvcnv  19861 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-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-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-dv 19754
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