Theorem f1osn 2827

Description: A singleton of an ordered pair is one-to-one onto function.

Hypotheses

Ref	Expression
f1osn.1	|- A e. V
f1osn.2	|- B e. V

Assertion

Ref	Expression
f1osn	|- {<.A, B>.}:{A}-1-1-onto->{B}

Proof of Theorem f1osn

Step	Hyp	Ref	Expression
1		f1osn.1	. . . . . 6 |- A e. V
2		f1osn.2	. . . . . 6 |- B e. V
3	1, 2	funsn 2690	. . . . 5 |- Fun {<.A, B>.}
4		dmsnop 2547	. . . . 5 |- dom {<.A, B>.} = {A}
5	3, 4	pm3.2i 234	. . . 4 |- (Fun {<.A, B>.} /\ dom {<.A, B>.} = {A})
6		df-fn 2433	. . . 4 |- ({<.A, B>.} Fn {A} <-> (Fun {<.A, B>.} /\ dom {<.A, B>.} = {A}))
7	5, 6	mpbir 165	. . 3 |- {<.A, B>.} Fn {A}
8	2, 1	funsn 2690	. . . . . 6 |- Fun {<.B, A>.}
9		dmsnop 2547	. . . . . 6 |- dom {<.B, A>.} = {B}
10	8, 9	pm3.2i 234	. . . . 5 |- (Fun {<.B, A>.} /\ dom {<.B, A>.} = {B})
11		df-fn 2433	. . . . 5 |- ({<.B, A>.} Fn {B} <-> (Fun {<.B, A>.} /\ dom {<.B, A>.} = {B}))
12	10, 11	mpbir 165	. . . 4 |- {<.B, A>.} Fn {B}
13	1, 2	cnvsn 2636	. . . . 5 |- `'{<.A, B>.} = {<.B, A>.}
14		fneq1 2718	. . . . 5 |- (`'{<.A, B>.} = {<.B, A>.} -> (`'{<.A, B>.} Fn {B} <-> {<.B, A>.} Fn {B}))
15	13, 14	ax-mp 6	. . . 4 |- (`'{<.A, B>.} Fn {B} <-> {<.B, A>.} Fn {B})
16	12, 15	mpbir 165	. . 3 |- `'{<.A, B>.} Fn {B}
17	7, 16	pm3.2i 234	. 2 |- ({<.A, B>.} Fn {A} /\ `'{<.A, B>.} Fn {B})
18		f1o4 2807	. 2 |- ({<.A, B>.}:{A}-1-1-onto->{B} <-> ({<.A, B>.} Fn {A} /\ `'{<.A, B>.} Fn {B}))
19	17, 18	mpbir 165	1 |- {<.A, B>.}:{A}-1-1-onto->{B}

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091   e. wcel 1092  Vcvv 1348  {csn 1808  <.cop 1810  `'ccnv 2409  dom cdm 2410  Fun wfun 2416   Fn wfn 2417  -1-1-onto->wf1o 2421

This theorem is referenced by:  fsn 2895  mapsn 3269  ensn1 3329  phplem3 3405  pssnn 3428  facnnt 4870  ruclem6 4890

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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