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Theorem fnsnb 6090
 Description: A function whose domain is a singleton can be represented as a singleton of an ordered pair. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) Revised to add reverse implication. (Revised by NM, 29-Dec-2018.)
Hypothesis
Ref Expression
fnsnb.1
Assertion
Ref Expression
fnsnb

Proof of Theorem fnsnb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 fnresdm 5695 . . . . . . . 8
2 fnfun 5683 . . . . . . . . 9
3 funressn 6084 . . . . . . . . 9
42, 3syl 16 . . . . . . . 8
51, 4eqsstr3d 3538 . . . . . . 7
65sseld 3502 . . . . . 6
7 elsni 4054 . . . . . 6
86, 7syl6 33 . . . . 5
9 df-fn 5596 . . . . . . . 8
10 fnsnb.1 . . . . . . . . . . 11
1110snid 4057 . . . . . . . . . 10
12 eleq2 2530 . . . . . . . . . 10
1311, 12mpbiri 233 . . . . . . . . 9
1413anim2i 569 . . . . . . . 8
159, 14sylbi 195 . . . . . . 7
16 funfvop 5999 . . . . . . 7
1715, 16syl 16 . . . . . 6
18 eleq1 2529 . . . . . 6
1917, 18syl5ibrcom 222 . . . . 5
208, 19impbid 191 . . . 4
21 elsn 4043 . . . 4
2220, 21syl6bbr 263 . . 3
2322eqrdv 2454 . 2
24 fvex 5881 . . . 4
2510, 24fnsn 5646 . . 3
26 fneq1 5674 . . 3
2725, 26mpbiri 233 . 2
2823, 27impbii 188 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818   cvv 3109  C_wss 3475  {csn 4029  <.cop 4035  domcdm 5004  |cres 5006  Funwfun 5587  Fnwfn 5588  cfv 5593 This theorem is referenced by:  fnprb  6129 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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 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-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  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-br 4453  df-opab 4511  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-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601
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