Theorem funprg 4466

Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)

Assertion

Ref	Expression
funprg	|- ((A =/= B /\ (A e. E /\ C e. F) /\ (B e. G /\ D e. H)) -> Fun {<.A, C>., <.B, D>.})

Proof of Theorem funprg

Step	Hyp	Ref	Expression
1		elisset 2299	. . . . . . 7 |- (A e. E -> A e. _V)
2		elisset 2299	. . . . . . 7 |- (C e. F -> C e. _V)
3	1, 2	anim12i 360	. . . . . 6 |- ((A e. E /\ C e. F) -> (A e. _V /\ C e. _V))
4		funsng 4465	. . . . . 6 |- ((A e. _V /\ C e. _V) -> Fun {<.A, C>.})
5	3, 4	syl 12	. . . . 5 |- ((A e. E /\ C e. F) -> Fun {<.A, C>.})
6		elisset 2299	. . . . . . 7 |- (B e. G -> B e. _V)
7		elisset 2299	. . . . . . 7 |- (D e. H -> D e. _V)
8	6, 7	anim12i 360	. . . . . 6 |- ((B e. G /\ D e. H) -> (B e. _V /\ D e. _V))
9		funsng 4465	. . . . . 6 |- ((B e. _V /\ D e. _V) -> Fun {<.B, D>.})
10	8, 9	syl 12	. . . . 5 |- ((B e. G /\ D e. H) -> Fun {<.B, D>.})
11	5, 10	anim12i 360	. . . 4 |- (((A e. E /\ C e. F) /\ (B e. G /\ D e. H)) -> (Fun {<.A, C>.} /\ Fun {<.B, D>.}))
12	11	3adant1 894	. . 3 |- ((A =/= B /\ (A e. E /\ C e. F) /\ (B e. G /\ D e. H)) -> (Fun {<.A, C>.} /\ Fun {<.B, D>.}))
13		dmsnop 4367	. . . . . . 7 |- dom {<.A, C>.} = {A}
14	13	a1i 8	. . . . . 6 |- (A =/= B -> dom {<.A, C>.} = {A})
15		dmsnop 4367	. . . . . . 7 |- dom {<.B, D>.} = {B}
16	15	a1i 8	. . . . . 6 |- (A =/= B -> dom {<.B, D>.} = {B})
17		ineq12 2791	. . . . . 6 |- ((dom {<.A, C>.} = {A} /\ dom {<.B, D>.} = {B}) -> (dom {<.A, C>.} i^i dom {<.B, D>.}) = ({A} i^i {B}))
18	14, 16, 17	syl11anc 524	. . . . 5 |- (A =/= B -> (dom {<.A, C>.} i^i dom {<.B, D>.}) = ({A} i^i {B}))
19		disjsn2 3091	. . . . 5 |- (A =/= B -> ({A} i^i {B}) = (/))
20	18, 19	eqtrd 1925	. . . 4 |- (A =/= B -> (dom {<.A, C>.} i^i dom {<.B, D>.}) = (/))
21	20	3ad2ant1 897	. . 3 |- ((A =/= B /\ (A e. E /\ C e. F) /\ (B e. G /\ D e. H)) -> (dom {<.A, C>.} i^i dom {<.B, D>.}) = (/))
22		funun 4462	. . 3 |- (((Fun {<.A, C>.} /\ Fun {<.B, D>.}) /\ (dom {<.A, C>.} i^i dom {<.B, D>.}) = (/)) -> Fun ({<.A, C>.} u. {<.B, D>.}))
23	12, 21, 22	syl11anc 524	. 2 |- ((A =/= B /\ (A e. E /\ C e. F) /\ (B e. G /\ D e. H)) -> Fun ({<.A, C>.} u. {<.B, D>.}))
24		df-pr 3050	. . . 4 |- {<.A, C>., <.B, D>.} = ({<.A, C>.} u. {<.B, D>.})
25	24	funeqi 4442	. . 3 |- (Fun {<.A, C>., <.B, D>.} <-> Fun ({<.A, C>.} u. {<.B, D>.}))
26	25	a1i 8	. 2 |- ((A =/= B /\ (A e. E /\ C e. F) /\ (B e. G /\ D e. H)) -> (Fun {<.A, C>., <.B, D>.} <-> Fun ({<.A, C>.} u. {<.B, D>.})))
27	23, 26	mpbird 213	1 |- ((A =/= B /\ (A e. E /\ C e. F) /\ (B e. G /\ D e. H)) -> Fun {<.A, C>., <.B, D>.})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044  {cpr 3045  <.cop 3046  dom cdm 3986  Fun wfun 3992

This theorem is referenced by:  funpr 4467  fnprg 4470

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-fun 4008

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