Theorem dmsnopOLD 4368

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

Assertion

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

Proof of Theorem dmsnopOLD

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . . . . 9 |- x e. _V
2		visset 2295	. . . . . . . . 9 |- y e. _V
3	1, 2	opthg 3533	. . . . . . . 8 |- (B e. _V -> (<.x, y>. = <.A, B>. <-> (x = A /\ y = B)))
4		opex 3527	. . . . . . . . 9 |- <.x, y>. e. _V
5	4	elsnc 3065	. . . . . . . 8 |- (<.x, y>. e. {<.A, B>.} <-> <.x, y>. = <.A, B>.)
6	3, 5	syl5bb 591	. . . . . . 7 |- (B e. _V -> (<.x, y>. e. {<.A, B>.} <-> (x = A /\ y = B)))
7	6	exbidv 1657	. . . . . 6 |- (B e. _V -> (E.y<.x, y>. e. {<.A, B>.} <-> E.y(x = A /\ y = B)))
8		19.42v 1688	. . . . . 6 |- (E.y(x = A /\ y = B) <-> (x = A /\ E.y y = B))
9	7, 8	syl6bb 595	. . . . 5 |- (B e. _V -> (E.y<.x, y>. e. {<.A, B>.} <-> (x = A /\ E.y y = B)))
10		isset 2296	. . . . . 6 |- (B e. _V <-> E.y y = B)
11		iba 704	. . . . . 6 |- (E.y y = B -> (x = A <-> (x = A /\ E.y y = B)))
12	10, 11	sylbi 216	. . . . 5 |- (B e. _V -> (x = A <-> (x = A /\ E.y y = B)))
13	9, 12	bitr4d 590	. . . 4 |- (B e. _V -> (E.y<.x, y>. e. {<.A, B>.} <-> x = A))
14	13	abbidv 2008	. . 3 |- (B e. _V -> {x | E.y<.x, y>. e. {<.A, B>.}} = {x | x = A})
15		dfdm3 4148	. . 3 |- dom {<.A, B>.} = {x | E.y<.x, y>. e. {<.A, B>.}}
16		df-sn 3049	. . 3 |- {A} = {x | x = A}
17	14, 15, 16	3eqtr4g 1953	. 2 |- (B e. _V -> dom {<.A, B>.} = {A})
18		opprc2 3171	. . . 4 |- (-. B e. _V -> <.A, B>. = <.A, A>.)
19		sneq 3054	. . . 4 |- (<.A, B>. = <.A, A>. -> {<.A, B>.} = {<.A, A>.})
20		dmeq 4157	. . . 4 |- ({<.A, B>.} = {<.A, A>.} -> dom {<.A, B>.} = dom {<.A, A>.})
21	18, 19, 20	3syl 24	. . 3 |- (-. B e. _V -> dom {<.A, B>.} = dom {<.A, A>.})
22	1, 2	opthg 3533	. . . . . . . . . 10 |- (A e. _V -> (<.x, y>. = <.A, A>. <-> (x = A /\ y = A)))
23	4	elsnc 3065	. . . . . . . . . 10 |- (<.x, y>. e. {<.A, A>.} <-> <.x, y>. = <.A, A>.)
24	22, 23	syl5bb 591	. . . . . . . . 9 |- (A e. _V -> (<.x, y>. e. {<.A, A>.} <-> (x = A /\ y = A)))
25	24	exbidv 1657	. . . . . . . 8 |- (A e. _V -> (E.y<.x, y>. e. {<.A, A>.} <-> E.y(x = A /\ y = A)))
26		19.42v 1688	. . . . . . . 8 |- (E.y(x = A /\ y = A) <-> (x = A /\ E.y y = A))
27	25, 26	syl6bb 595	. . . . . . 7 |- (A e. _V -> (E.y<.x, y>. e. {<.A, A>.} <-> (x = A /\ E.y y = A)))
28		isset 2296	. . . . . . . 8 |- (A e. _V <-> E.y y = A)
29		iba 704	. . . . . . . 8 |- (E.y y = A -> (x = A <-> (x = A /\ E.y y = A)))
30	28, 29	sylbi 216	. . . . . . 7 |- (A e. _V -> (x = A <-> (x = A /\ E.y y = A)))
31	27, 30	bitr4d 590	. . . . . 6 |- (A e. _V -> (E.y<.x, y>. e. {<.A, A>.} <-> x = A))
32	31	abbidv 2008	. . . . 5 |- (A e. _V -> {x | E.y<.x, y>. e. {<.A, A>.}} = {x | x = A})
33		dfdm3 4148	. . . . 5 |- dom {<.A, A>.} = {x | E.y<.x, y>. e. {<.A, A>.}}
34	32, 33, 16	3eqtr4g 1953	. . . 4 |- (A e. _V -> dom {<.A, A>.} = {A})
35		opprc1b 3542	. . . . . 6 |- (-. A e. _V <-> (/) e. <.A, A>.)
36		dmsn0el 4366	. . . . . 6 |- ((/) e. <.A, A>. -> dom {<.A, A>.} = (/))
37	35, 36	sylbi 216	. . . . 5 |- (-. A e. _V -> dom {<.A, A>.} = (/))
38		snprc 3092	. . . . . 6 |- (-. A e. _V <-> {A} = (/))
39	38	biimpi 168	. . . . 5 |- (-. A e. _V -> {A} = (/))
40	37, 39	eqtr4d 1928	. . . 4 |- (-. A e. _V -> dom {<.A, A>.} = {A})
41	34, 40	pm2.61i 140	. . 3 |- dom {<.A, A>.} = {A}
42	21, 41	syl6eq 1944	. 2 |- (-. B e. _V -> dom {<.A, B>.} = {A})
43	17, 42	pm2.61i 140	1 |- dom {<.A, B>.} = {A}

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  _Vcvv 2292  (/)c0 2875  {csn 3044  <.cop 3046  dom cdm 3986

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-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  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-xp 4000  df-dm 4004

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