Theorem f1osn 3719

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		f1o4 3696	. 2 |- ({<.A, B>.}:{A}-1-1-onto->{B} <-> ({<.A, B>.} Fn {A} /\ `'{<.A, B>.} Fn {B}))
2		df-fn 3193	. . 3 |- ({<.A, B>.} Fn {A} <-> (Fun {<.A, B>.} /\ dom {<.A, B>.} = {A}))
3		f1osn.1	. . . 4 |- A e. V
4		f1osn.2	. . . 4 |- B e. V
5	3, 4	funsn 3543	. . 3 |- Fun {<.A, B>.}
6		dmsnop 3328	. . 3 |- dom {<.A, B>.} = {A}
7	2, 5, 6	mpbir2an 730	. 2 |- {<.A, B>.} Fn {A}
8		df-fn 3193	. . . 4 |- ({<.B, A>.} Fn {B} <-> (Fun {<.B, A>.} /\ dom {<.B, A>.} = {B}))
9	4, 3	funsn 3543	. . . 4 |- Fun {<.B, A>.}
10		dmsnop 3328	. . . 4 |- dom {<.B, A>.} = {B}
11	8, 9, 10	mpbir2an 730	. . 3 |- {<.B, A>.} Fn {B}
12	3, 4	cnvsn 3449	. . . 4 |- `'{<.A, B>.} = {<.B, A>.}
13		fneq1 3582	. . . 4 |- (`'{<.A, B>.} = {<.B, A>.} -> (`'{<.A, B>.} Fn {B} <-> {<.B, A>.} Fn {B}))
14	12, 13	ax-mp 7	. . 3 |- (`'{<.A, B>.} Fn {B} <-> {<.B, A>.} Fn {B})
15	11, 14	mpbir 190	. 2 |- `'{<.A, B>.} Fn {B}
16	1, 7, 15	mpbir2an 730	1 |- {<.A, B>.}:{A}-1-1-onto->{B}

Syntax hints:   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  <.cop 2411  `'ccnv 3169  dom cdm 3170  Fun wfun 3176   Fn wfn 3177  -1-1-onto->wf1o 3181

This theorem is referenced by:  fvsnun2 3796  fsn 3834  fopabsn 3840  mapsn 4345  ensn1 4424  phplem2 4509  pssnn 4534  acdc2lem2 7489  acdc5lem2 7492  ruclem6 7515  grpsn 8124  1alg 10654

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

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