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Theorem dfrel4 23987
 Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5731 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
Hypotheses
Ref Expression
dfrel4.1
dfrel4.2
Assertion
Ref Expression
dfrel4
Distinct variable group:   ,
Allowed substitution hints:   (,)

Proof of Theorem dfrel4
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrel4v 5281 . 2
2 nfcv 2540 . . . . 5
3 dfrel4.1 . . . . 5
4 nfcv 2540 . . . . 5
52, 3, 4nfbr 4216 . . . 4
6 nfcv 2540 . . . . 5
7 dfrel4.2 . . . . 5
8 nfcv 2540 . . . . 5
96, 7, 8nfbr 4216 . . . 4
10 nfv 1626 . . . 4
11 nfv 1626 . . . 4
12 breq12 4177 . . . 4
135, 9, 10, 11, 12cbvopab 4236 . . 3
1413eqeq2i 2414 . 2
151, 14bitri 241 1
 Colors of variables: wff set class Syntax hints:   wb 177   wceq 1649  wnfc 2527   class class class wbr 4172  copab 4225   wrel 4842 This theorem is referenced by:  feqmptdf  24028 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-xp 4843  df-rel 4844  df-cnv 4845
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