Theorem dmsnop 2547

Description: The domain of a singleton of an ordered pair is the singleton of the first member.

Assertion

Ref	Expression
dmsnop	|- dom {<.A, B>.} = {A}

Proof of Theorem dmsnop

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . . 9 |- x e. V
2		visset 1350	. . . . . . . . 9 |- y e. V
3	1, 2	opthg 1899	. . . . . . . 8 |- (B e. V -> (<.x, y>. = <.A, B>. <-> (x = A /\ y = B)))
4		opex 1893	. . . . . . . . 9 |- <.x, y>. e. V
5	4	elsnc 1826	. . . . . . . 8 |- (<.x, y>. e. {<.A, B>.} <-> <.x, y>. = <.A, B>.)
6	3, 5	syl5bb 410	. . . . . . 7 |- (B e. V -> (<.x, y>. e. {<.A, B>.} <-> (x = A /\ y = B)))
7	6	biexdv 936	. . . . . 6 |- (B e. V -> (E.y<.x, y>. e. {<.A, B>.} <-> E.y(x = A /\ y = B)))
8		19.42v 966	. . . . . 6 |- (E.y(x = A /\ y = B) <-> (x = A /\ E.y y = B))
9	7, 8	syl6bb 414	. . . . 5 |- (B e. V -> (E.y<.x, y>. e. {<.A, B>.} <-> (x = A /\ E.y y = B)))
10		isset 1351	. . . . . 6 |- (B e. V <-> E.y y = B)
11		iba 486	. . . . . 6 |- (E.y y = B -> (x = A <-> (x = A /\ E.y y = B)))
12	10, 11	sylbi 174	. . . . 5 |- (B e. V -> (x = A <-> (x = A /\ E.y y = B)))
13	9, 12	bitr4d 409	. . . 4 |- (B e. V -> (E.y<.x, y>. e. {<.A, B>.} <-> x = A))
14	13	biabdv 1183	. . 3 |- (B e. V -> {x | E.y<.x, y>. e. {<.A, B>.}} = {x | x = A})
15		dfdm3 2522	. . 3 |- dom {<.A, B>.} = {x | E.y<.x, y>. e. {<.A, B>.}}
16		df-sn 1811	. . 3 |- {A} = {x | x = A}
17	14, 15, 16	3eqtr4g 1147	. 2 |- (B e. V -> dom {<.A, B>.} = {A})
18		opprc2 1907	. . . 4 |- (-. B e. V -> <.A, B>. = <.A, A>.)
19		sneq 1816	. . . 4 |- (<.A, B>. = <.A, A>. -> {<.A, B>.} = {<.A, A>.})
20		dmeq 2531	. . . 4 |- ({<.A, B>.} = {<.A, A>.} -> dom {<.A, B>.} = dom {<.A, A>.})
21	18, 19, 20	3syl 21	. . 3 |- (-. B e. V -> dom {<.A, B>.} = dom {<.A, A>.})
22	1, 2	opthg 1899	. . . . . . . . . 10 |- (A e. V -> (<.x, y>. = <.A, A>. <-> (x = A /\ y = A)))
23	4	elsnc 1826	. . . . . . . . . 10 |- (<.x, y>. e. {<.A, A>.} <-> <.x, y>. = <.A, A>.)
24	22, 23	syl5bb 410	. . . . . . . . 9 |- (A e. V -> (<.x, y>. e. {<.A, A>.} <-> (x = A /\ y = A)))
25	24	biexdv 936	. . . . . . . 8 |- (A e. V -> (E.y<.x, y>. e. {<.A, A>.} <-> E.y(x = A /\ y = A)))
26		19.42v 966	. . . . . . . 8 |- (E.y(x = A /\ y = A) <-> (x = A /\ E.y y = A))
27	25, 26	syl6bb 414	. . . . . . 7 |- (A e. V -> (E.y<.x, y>. e. {<.A, A>.} <-> (x = A /\ E.y y = A)))
28		isset 1351	. . . . . . . 8 |- (A e. V <-> E.y y = A)
29		iba 486	. . . . . . . 8 |- (E.y y = A -> (x = A <-> (x = A /\ E.y y = A)))
30	28, 29	sylbi 174	. . . . . . 7 |- (A e. V -> (x = A <-> (x = A /\ E.y y = A)))
31	27, 30	bitr4d 409	. . . . . 6 |- (A e. V -> (E.y<.x, y>. e. {<.A, A>.} <-> x = A))
32	31	biabdv 1183	. . . . 5 |- (A e. V -> {x | E.y<.x, y>. e. {<.A, A>.}} = {x | x = A})
33		dfdm3 2522	. . . . 5 |- dom {<.A, A>.} = {x | E.y<.x, y>. e. {<.A, A>.}}
34	32, 33, 16	3eqtr4g 1147	. . . 4 |- (A e. V -> dom {<.A, A>.} = {A})
35		anidm 331	. . . . . . . 8 |- ((-. A e. V /\ -. A e. V) <-> -. A e. V)
36		opprc3 1908	. . . . . . . 8 |- ((-. A e. V /\ -. A e. V) <-> <.A, A>. = {(/)})
37	35, 36	bitr3 153	. . . . . . 7 |- (-. A e. V <-> <.A, A>. = {(/)})
38		sneq 1816	. . . . . . . 8 |- (<.A, A>. = {(/)} -> {<.A, A>.} = {{(/)}})
39	38	dmeqd 2533	. . . . . . 7 |- (<.A, A>. = {(/)} -> dom {<.A, A>.} = dom {{(/)}})
40	37, 39	sylbi 174	. . . . . 6 |- (-. A e. V -> dom {<.A, A>.} = dom {{(/)}})
41		dmsnsn0 2544	. . . . . 6 |- dom {{(/)}} = (/)
42	40, 41	syl6eq 1140	. . . . 5 |- (-. A e. V -> dom {<.A, A>.} = (/))
43		snprc 1838	. . . . . 6 |- (-. A e. V <-> {A} = (/))
44	43	biimp 133	. . . . 5 |- (-. A e. V -> {A} = (/))
45	42, 44	eqtr4d 1131	. . . 4 |- (-. A e. V -> dom {<.A, A>.} = {A})
46	34, 45	pm2.61i 110	. . 3 |- dom {<.A, A>.} = {A}
47	21, 46	syl6eq 1140	. 2 |- (-. B e. V -> dom {<.A, B>.} = {A})
48	17, 47	pm2.61i 110	1 |- dom {<.A, B>.} = {A}

Syntax hints:  -. wn 1   <-> wb 127   /\ wa 196  E.wex 678  {cab 1090   = wceq 1091   e. wcel 1092  Vcvv 1348  (/)c0 1707  {csn 1808  <.cop 1810  dom cdm 2410

This theorem is referenced by:  dmsnsnsn 2548  op1sta 2635  rnsnop 2637  f1osn 2827  tfrlem10 2958  fac0 4871

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-ex 679  df-sb 853  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-dm 2428

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