Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  relmptopab Structured version   Unicode version

Theorem relmptopab 6294
 Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 7-Aug-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
relmptopab.1
Assertion
Ref Expression
relmptopab
Distinct variable group:   ,
Allowed substitution hints:   (,,)   (,)   (,,)   (,,)

Proof of Theorem relmptopab
StepHypRef Expression
1 relmptopab.1 . . . 4
21fvmptss 5815 . . 3
3 relopab 5003 . . . . 5
4 df-rel 4887 . . . . 5
53, 4mpbi 201 . . . 4
65a1i 11 . . 3
72, 6mprg 2777 . 2
8 df-rel 4887 . 2
97, 8mpbir 202 1
 Colors of variables: wff set class Syntax hints:   wceq 1653   wcel 1726  cvv 2958   wss 3322  copab 4267   cmpt 4268   cxp 4878   wrel 4885  cfv 5456 This theorem is referenced by:  reldvdsr  15751  lmrel  17296  phtpcrel  19020  ulmrel  20296 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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405 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 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464
 Copyright terms: Public domain W3C validator