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Theorem fnsnb 6081
 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 5690 . . . . . . . . 9
2 fnfun 5678 . . . . . . . . . 10
3 funressn 6075 . . . . . . . . . 10
42, 3syl 16 . . . . . . . . 9
51, 4eqsstr3d 3539 . . . . . . . 8
65sseld 3503 . . . . . . 7
7 elsni 4052 . . . . . . 7
86, 7syl6 33 . . . . . 6
9 df-fn 5591 . . . . . . . . 9
10 fnsnb.1 . . . . . . . . . . . 12
1110snid 4055 . . . . . . . . . . 11
12 eleq2 2540 . . . . . . . . . . 11
1311, 12mpbiri 233 . . . . . . . . . 10
1413anim2i 569 . . . . . . . . 9
159, 14sylbi 195 . . . . . . . 8
16 funfvop 5994 . . . . . . . 8
1715, 16syl 16 . . . . . . 7
18 eleq1 2539 . . . . . . 7
1917, 18syl5ibrcom 222 . . . . . 6
208, 19impbid 191 . . . . 5
21 elsn 4041 . . . . . 6
2221bibi2i 313 . . . . 5
2320, 22sylibr 212 . . . 4
2423alrimiv 1695 . . 3
25 dfcleq 2460 . . 3
2624, 25sylibr 212 . 2
27 fvex 5876 . . . 4
2810, 27fnsn 5641 . . 3
29 fneq1 5669 . . 3
3028, 29mpbiri 233 . 2
3126, 30impbii 188 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369  wal 1377   wceq 1379   wcel 1767  cvv 3113   wss 3476  csn 4027  cop 4033   cdm 4999   cres 5001   wfun 5582   wfn 5583  cfv 5588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596 This theorem is referenced by:  fnprb  6120
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