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Theorem relmpt2opab 6882
Description: Any function to sets of ordered pairs produces a relation on function value unconditionally. (Contributed by Mario Carneiro, 9-Feb-2015.)
Hypothesis
Ref Expression
relmpt2opab.1
Assertion
Ref Expression
relmpt2opab
Distinct variable groups:   , , ,   ,   , ,

Proof of Theorem relmpt2opab
StepHypRef Expression
1 relopab 5134 . . . . 5
2 df-rel 5011 . . . . 5
31, 2mpbi 208 . . . 4
43rgen2w 2819 . . 3
5 relmpt2opab.1 . . . 4
65ovmptss 6881 . . 3
74, 6ax-mp 5 . 2
8 df-rel 5011 . 2
97, 8mpbir 209 1
Colors of variables: wff setvar class
Syntax hints:  =wceq 1395  A.wral 2807   cvv 3109  C_wss 3475  {copab 4509  X.cxp 5002  Relwrel 5009  (class class class)co 6296  e.cmpt2 6298
This theorem is referenced by:  brovmpt2ex  6970  relfunc  15231  releqg  16248  relwlk  24518  releupa  24964
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801
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