Theorem fprOLD 4811

Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.)

Hypotheses

Ref	Expression
fpr.1	|- A e. _V
fpr.2	|- B e. _V
fpr.3	|- C e. _V
fpr.4	|- D e. _V

Assertion

Ref	Expression
fprOLD	|- (A =/= B -> {<.A, C>., <.B, D>.}:{A, B}-->{C, D})

Proof of Theorem fprOLD

Step	Hyp	Ref	Expression
1		df-f 4010	. 2 |- ({<.A, C>., <.B, D>.}:{A, B}-->{C, D} <-> ({<.A, C>., <.B, D>.} Fn {A, B} /\ ran {<.A, C>., <.B, D>.} C_ {C, D}))
2		fpr.1	. . 3 |- A e. _V
3		fpr.3	. . 3 |- C e. _V
4		fpr.2	. . . . 5 |- B e. _V
5		fpr.4	. . . . 5 |- D e. _V
6	4, 5	pm3.2i 307	. . . 4 |- (B e. _V /\ D e. _V)
7		fnprg 4470	. . . 4 |- ((A =/= B /\ (A e. _V /\ C e. _V) /\ (B e. _V /\ D e. _V)) -> {<.A, C>., <.B, D>.} Fn {A, B})
8	6, 7	mp3an3 1180	. . 3 |- ((A =/= B /\ (A e. _V /\ C e. _V)) -> {<.A, C>., <.B, D>.} Fn {A, B})
9	2, 3, 8	mpanr12 778	. 2 |- (A =/= B -> {<.A, C>., <.B, D>.} Fn {A, B})
10		df-pr 3050	. . . . . 6 |- {<.A, C>., <.B, D>.} = ({<.A, C>.} u. {<.B, D>.})
11	10	rneqi 4187	. . . . 5 |- ran {<.A, C>., <.B, D>.} = ran ({<.A, C>.} u. {<.B, D>.})
12		rnun 4325	. . . . 5 |- ran ({<.A, C>.} u. {<.B, D>.}) = (ran {<.A, C>.} u. ran {<.B, D>.})
13	2, 3	rnsnop 4375	. . . . . . 7 |- ran {<.A, C>.} = {C}
14	4, 5	rnsnop 4375	. . . . . . 7 |- ran {<.B, D>.} = {D}
15	13, 14	uneq12i 2753	. . . . . 6 |- (ran {<.A, C>.} u. ran {<.B, D>.}) = ({C} u. {D})
16		df-pr 3050	. . . . . 6 |- {C, D} = ({C} u. {D})
17	15, 16	eqtr4i 1911	. . . . 5 |- (ran {<.A, C>.} u. ran {<.B, D>.}) = {C, D}
18	11, 12, 17	3eqtri 1912	. . . 4 |- ran {<.A, C>., <.B, D>.} = {C, D}
19	18	eqimssi 2668	. . 3 |- ran {<.A, C>., <.B, D>.} C_ {C, D}
20	19	a1i 8	. 2 |- (A =/= B -> ran {<.A, C>., <.B, D>.} C_ {C, D})
21	1, 9, 20	sylanbrc 527	1 |- (A =/= B -> {<.A, C>., <.B, D>.}:{A, B}-->{C, D})

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300   =/= wne 2017  _Vcvv 2292   u. cun 2591   C_ wss 2593  {csn 3044  {cpr 3045  <.cop 3046  ran crn 3987   Fn wfn 3993  -->wf 3994

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-fun 4008  df-fn 4009  df-f 4010

Copyright terms: Public domain