Theorem dmsnop 3285

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 1788	. . . . . . . . 9 |- x e. V
2		visset 1788	. . . . . . . . 9 |- y e. V
3	1, 2	opthg 2755	. . . . . . . 8 |- (B e. V -> (<.x, y>. = <.A, B>. <-> (x = A /\ y = B)))
4		opex 2750	. . . . . . . . 9 |- <.x, y>. e. V
5	4	elsnc 2402	. . . . . . . 8 |- (<.x, y>. e. {<.A, B>.} <-> <.x, y>. = <.A, B>.)
6	3, 5	syl5bb 530	. . . . . . 7 |- (B e. V -> (<.x, y>. e. {<.A, B>.} <-> (x = A /\ y = B)))
7	6	exbidv 1261	. . . . . 6 |- (B e. V -> (E.y<.x, y>. e. {<.A, B>.} <-> E.y(x = A /\ y = B)))
8		19.42v 1290	. . . . . 6 |- (E.y(x = A /\ y = B) <-> (x = A /\ E.y y = B))
9	7, 8	syl6bb 534	. . . . 5 |- (B e. V -> (E.y<.x, y>. e. {<.A, B>.} <-> (x = A /\ E.y y = B)))
10		isset 1789	. . . . . 6 |- (B e. V <-> E.y y = B)
11		iba 640	. . . . . 6 |- (E.y y = B -> (x = A <-> (x = A /\ E.y y = B)))
12	10, 11	sylbi 199	. . . . 5 |- (B e. V -> (x = A <-> (x = A /\ E.y y = B)))
13	9, 12	bitr4d 529	. . . 4 |- (B e. V -> (E.y<.x, y>. e. {<.A, B>.} <-> x = A))
14	13	abbidv 1553	. . 3 |- (B e. V -> {x | E.y<.x, y>. e. {<.A, B>.}} = {x | x = A})
15		dfdm3 3259	. . 3 |- dom {<.A, B>.} = {x | E.y<.x, y>. e. {<.A, B>.}}
16		df-sn 2383	. . 3 |- {A} = {x | x = A}
17	14, 15, 16	3eqtr4g 1507	. 2 |- (B e. V -> dom {<.A, B>.} = {A})
18		opprc2 2468	. . . 4 |- (-. B e. V -> <.A, B>. = <.A, A>.)
19		sneq 2388	. . . 4 |- (<.A, B>. = <.A, A>. -> {<.A, B>.} = {<.A, A>.})
20		dmeq 3268	. . . 4 |- ({<.A, B>.} = {<.A, A>.} -> dom {<.A, B>.} = dom {<.A, A>.})
21	18, 19, 20	3syl 20	. . 3 |- (-. B e. V -> dom {<.A, B>.} = dom {<.A, A>.})
22	1, 2	opthg 2755	. . . . . . . . . 10 |- (A e. V -> (<.x, y>. = <.A, A>. <-> (x = A /\ y = A)))
23	4	elsnc 2402	. . . . . . . . . 10 |- (<.x, y>. e. {<.A, A>.} <-> <.x, y>. = <.A, A>.)
24	22, 23	syl5bb 530	. . . . . . . . 9 |- (A e. V -> (<.x, y>. e. {<.A, A>.} <-> (x = A /\ y = A)))
25	24	exbidv 1261	. . . . . . . 8 |- (A e. V -> (E.y<.x, y>. e. {<.A, A>.} <-> E.y(x = A /\ y = A)))
26		19.42v 1290	. . . . . . . 8 |- (E.y(x = A /\ y = A) <-> (x = A /\ E.y y = A))
27	25, 26	syl6bb 534	. . . . . . 7 |- (A e. V -> (E.y<.x, y>. e. {<.A, A>.} <-> (x = A /\ E.y y = A)))
28		isset 1789	. . . . . . . 8 |- (A e. V <-> E.y y = A)
29		iba 640	. . . . . . . 8 |- (E.y y = A -> (x = A <-> (x = A /\ E.y y = A)))
30	28, 29	sylbi 199	. . . . . . 7 |- (A e. V -> (x = A <-> (x = A /\ E.y y = A)))
31	27, 30	bitr4d 529	. . . . . 6 |- (A e. V -> (E.y<.x, y>. e. {<.A, A>.} <-> x = A))
32	31	abbidv 1553	. . . . 5 |- (A e. V -> {x | E.y<.x, y>. e. {<.A, A>.}} = {x | x = A})
33		dfdm3 3259	. . . . 5 |- dom {<.A, A>.} = {x | E.y<.x, y>. e. {<.A, A>.}}
34	32, 33, 16	3eqtr4g 1507	. . . 4 |- (A e. V -> dom {<.A, A>.} = {A})
35		dmsnsn0 3282	. . . . 5 |- dom {{(/)}} = (/)
36		anidm 432	. . . . . . 7 |- ((-. A e. V /\ -. A e. V) <-> -. A e. V)
37		opprc3 2764	. . . . . . 7 |- ((-. A e. V /\ -. A e. V) <-> <.A, A>. = {(/)})
38	36, 37	bitr3 175	. . . . . 6 |- (-. A e. V <-> <.A, A>. = {(/)})
39		sneq 2388	. . . . . . 7 |- (<.A, A>. = {(/)} -> {<.A, A>.} = {{(/)}})
40	39	dmeqd 3270	. . . . . 6 |- (<.A, A>. = {(/)} -> dom {<.A, A>.} = dom {{(/)}})
41	38, 40	sylbi 199	. . . . 5 |- (-. A e. V -> dom {<.A, A>.} = dom {{(/)}})
42		snprc 2414	. . . . . 6 |- (-. A e. V <-> {A} = (/))
43	42	biimp 151	. . . . 5 |- (-. A e. V -> {A} = (/))
44	35, 41, 43	3eqtr4a 1508	. . . 4 |- (-. A e. V -> dom {<.A, A>.} = {A})
45	34, 44	pm2.61i 126	. . 3 |- dom {<.A, A>.} = {A}
46	21, 45	syl6eq 1499	. 2 |- (-. B e. V -> dom {<.A, B>.} = {A})
47	17, 46	pm2.61i 126	1 |- dom {<.A, B>.} = {A}

Syntax hints:  -. wn 2   <-> wb 146   /\ wa 223  E.wex 956   = wceq 1099   e. wcel 1105  {cab 1440  Vcvv 1786  (/)c0 2251  {csn 2380  <.cop 2382  dom cdm 3133

This theorem is referenced by:  dmsnsnsn 3286  op1sta 3397  rnsnop 3399  f1osn 3658  tfrlem10 3859  ringsn 8048  1alg 8848  1ded 8865  1cat 8886

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-br 2588  df-dm 3151

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