Theorem iunfopab 4542

Description: Two ways to express a function as a class of ordered pairs. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Sep-2011.)

Hypothesis

Ref	Expression
iunfopab.1	|- B e. _V

Assertion

Ref	Expression
iunfopab	|- U_x e. A {<.x, B>.} = {<.x, y>. | (x e. A /\ y = B)}

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem iunfopab

Step	Hyp	Ref	Expression
1		df-rex 2110	. . . 4 |- (E.x e. A z e. {<.x, B>.} <-> E.x(x e. A /\ z e. {<.x, B>.}))
2		elsn 3058	. . . . . . 7 |- (z e. {<.x, B>.} <-> z = <.x, B>.)
3	2	anbi2i 538	. . . . . 6 |- ((x e. A /\ z e. {<.x, B>.}) <-> (x e. A /\ z = <.x, B>.))
4		iunfopab.1	. . . . . . 7 |- B e. _V
5		opeq2 3159	. . . . . . . . 9 |- (y = B -> <.x, y>. = <.x, B>.)
6	5	eqeq2d 1895	. . . . . . . 8 |- (y = B -> (z = <.x, y>. <-> z = <.x, B>.))
7	6	anbi2d 678	. . . . . . 7 |- (y = B -> ((x e. A /\ z = <.x, y>.) <-> (x e. A /\ z = <.x, B>.)))
8	4, 7	ceqsexv 2325	. . . . . 6 |- (E.y(y = B /\ (x e. A /\ z = <.x, y>.)) <-> (x e. A /\ z = <.x, B>.))
9		3anrev 868	. . . . . . . 8 |- ((y = B /\ x e. A /\ z = <.x, y>.) <-> (z = <.x, y>. /\ x e. A /\ y = B))
10		3anass 862	. . . . . . . 8 |- ((y = B /\ x e. A /\ z = <.x, y>.) <-> (y = B /\ (x e. A /\ z = <.x, y>.)))
11		3anass 862	. . . . . . . 8 |- ((z = <.x, y>. /\ x e. A /\ y = B) <-> (z = <.x, y>. /\ (x e. A /\ y = B)))
12	9, 10, 11	3bitr3i 198	. . . . . . 7 |- ((y = B /\ (x e. A /\ z = <.x, y>.)) <-> (z = <.x, y>. /\ (x e. A /\ y = B)))
13	12	exbii 1398	. . . . . 6 |- (E.y(y = B /\ (x e. A /\ z = <.x, y>.)) <-> E.y(z = <.x, y>. /\ (x e. A /\ y = B)))
14	3, 8, 13	3bitr2i 196	. . . . 5 |- ((x e. A /\ z e. {<.x, B>.}) <-> E.y(z = <.x, y>. /\ (x e. A /\ y = B)))
15	14	exbii 1398	. . . 4 |- (E.x(x e. A /\ z e. {<.x, B>.}) <-> E.xE.y(z = <.x, y>. /\ (x e. A /\ y = B)))
16	1, 15	bitri 190	. . 3 |- (E.x e. A z e. {<.x, B>.} <-> E.xE.y(z = <.x, y>. /\ (x e. A /\ y = B)))
17	16	abbii 2006	. 2 |- {z | E.x e. A z e. {<.x, B>.}} = {z | E.xE.y(z = <.x, y>. /\ (x e. A /\ y = B))}
18		df-iun 3257	. 2 |- U_x e. A {<.x, B>.} = {z | E.x e. A z e. {<.x, B>.}}
19		df-opab 3396	. 2 |- {<.x, y>. | (x e. A /\ y = B)} = {z | E.xE.y(z = <.x, y>. /\ (x e. A /\ y = B))}
20	17, 18, 19	3eqtr4i 1921	1 |- U_x e. A {<.x, B>.} = {<.x, y>. | (x e. A /\ y = B)}

Syntax hints:   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  E.wrex 2106  _Vcvv 2292  {csn 3044  <.cop 3046  U_ciun 3255  {copab 3395

This theorem is referenced by:  iunfoprab 5072  dfmpt 5073

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-rex 2110  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-op 3053  df-iun 3257  df-opab 3396

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