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Theorem fgraphxp 36159
 Description: Express a function as a subset of the Cartesian product. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Assertion
Ref Expression
fgraphxp
Distinct variable groups:   ,   ,   ,

Proof of Theorem fgraphxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fgraphopab 36158 . 2
2 vex 3034 . . . . . . 7
3 vex 3034 . . . . . . 7
42, 3op1std 6822 . . . . . 6
54fveq2d 5883 . . . . 5
62, 3op2ndd 6823 . . . . 5
75, 6eqeq12d 2486 . . . 4
87rabxp 4876 . . 3
9 df-3an 1009 . . . 4
109opabbii 4460 . . 3
118, 10eqtri 2493 . 2
121, 11syl6eqr 2523 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   w3a 1007   wceq 1452   wcel 1904  crab 2760  cop 3965  copab 4453   cxp 4837  wf 5585  cfv 5589  c1st 6810  c2nd 6811 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-fv 5597  df-1st 6812  df-2nd 6813 This theorem is referenced by:  hausgraph  36160
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