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Theorem fnsingle 25764
 Description: The singleton relationship is a function over the universe. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fnsingle Singleton

Proof of Theorem fnsingle
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difss 3474 . . . . 5 (++)
2 df-rel 4885 . . . . 5 (++) (++)
31, 2mpbir 201 . . . 4 (++)
4 df-singleton 25706 . . . . 5 Singleton (++)
54releqi 4960 . . . 4 Singleton (++)
63, 5mpbir 201 . . 3 Singleton
7 vex 2959 . . . . . . 7
8 vex 2959 . . . . . . 7
97, 8brsingle 25762 . . . . . 6 Singleton
10 vex 2959 . . . . . . 7
117, 10brsingle 25762 . . . . . 6 Singleton
12 eqtr3 2455 . . . . . 6
139, 11, 12syl2anb 466 . . . . 5 Singleton Singleton
1413ax-gen 1555 . . . 4 Singleton Singleton
1514gen2 1556 . . 3 Singleton Singleton
16 dffun2 5464 . . 3 Singleton Singleton Singleton Singleton
176, 15, 16mpbir2an 887 . 2 Singleton
18 eqv 3643 . . 3 Singleton Singleton
19 eqid 2436 . . . . . 6
20 snex 4405 . . . . . . 7
217, 20brsingle 25762 . . . . . 6 Singleton
2219, 21mpbir 201 . . . . 5 Singleton
23 breq2 4216 . . . . . 6 Singleton Singleton
2420, 23spcev 3043 . . . . 5 Singleton Singleton
2522, 24ax-mp 8 . . . 4 Singleton
267eldm 5067 . . . 4 Singleton Singleton
2725, 26mpbir 201 . . 3 Singleton
2818, 27mpgbir 1559 . 2 Singleton
29 df-fn 5457 . 2 Singleton Singleton Singleton
3017, 28, 29mpbir2an 887 1 Singleton
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549  wex 1550   wceq 1652   wcel 1725  cvv 2956   cdif 3317   wss 3320  csn 3814   class class class wbr 4212   cep 4492   cid 4493   cxp 4876   cdm 4878   crn 4879   wrel 4883   wfun 5448   wfn 5449  (++)csymdif 25662   ctxp 25674  Singletoncsingle 25682 This theorem is referenced by:  fvsingle  25765 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-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-br 4213  df-opab 4267  df-mpt 4268  df-eprel 4494  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-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fo 5460  df-fv 5462  df-1st 6349  df-2nd 6350  df-symdif 25663  df-txp 25698  df-singleton 25706
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