Theorem iunfopabOLD 4543

Description: Two ways to express a function as a class of ordered pairs.

Hypothesis

Ref	Expression
iunfopab.1OLD	|- B e. _V

Assertion

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

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

Proof of Theorem iunfopabOLD

Step	Hyp	Ref	Expression
1		reliun 4101	. . 3 |- (Rel U_x e. A {<.x, B>.} <-> A.x e. A Rel {<.x, B>.})
2		visset 2295	. . . . 5 |- x e. _V
3	2	relsn 4087	. . . 4 |- Rel {<.x, B>.}
4	3	a1i 8	. . 3 |- (x e. A -> Rel {<.x, B>.})
5	1, 4	mprgbir 2163	. 2 |- Rel U_x e. A {<.x, B>.}
6		relopab 4104	. 2 |- Rel {<.x, y>. | (x e. A /\ y = B)}
7		visset 2295	. . . . . . 7 |- z e. _V
8		visset 2295	. . . . . . 7 |- w e. _V
9		iunfopab.1OLD	. . . . . . 7 |- B e. _V
10	7, 8, 9	opth 3532	. . . . . 6 |- (<.z, w>. = <.x, B>. <-> (z = x /\ w = B))
11		opex 3527	. . . . . . 7 |- <.z, w>. e. _V
12	11	elsnc 3065	. . . . . 6 |- (<.z, w>. e. {<.x, B>.} <-> <.z, w>. = <.x, B>.)
13		eqcom 1886	. . . . . . 7 |- (x = z <-> z = x)
14	13	anbi1i 539	. . . . . 6 |- ((x = z /\ w = B) <-> (z = x /\ w = B))
15	10, 12, 14	3bitr4i 200	. . . . 5 |- (<.z, w>. e. {<.x, B>.} <-> (x = z /\ w = B))
16	15	rexbii 2128	. . . 4 |- (E.x e. A <.z, w>. e. {<.x, B>.} <-> E.x e. A (x = z /\ w = B))
17		df-rex 2110	. . . . 5 |- (E.x e. A (x = z /\ w = B) <-> E.x(x e. A /\ (x = z /\ w = B)))
18		an12 542	. . . . . 6 |- ((x = z /\ (x e. A /\ w = B)) <-> (x e. A /\ (x = z /\ w = B)))
19	18	exbii 1398	. . . . 5 |- (E.x(x = z /\ (x e. A /\ w = B)) <-> E.x(x e. A /\ (x = z /\ w = B)))
20		ax-17 1317	. . . . . . 7 |- (z e. A -> A.x z e. A)
21		ax-17 1317	. . . . . . . . 9 |- (w e. z -> A.x w e. z)
22	7, 21	hbcsb1 2568	. . . . . . . 8 |- (w e. [_z / x]_B -> A.x w e. [_z / x]_B)
23	22	hbeleq 1997	. . . . . . 7 |- (w = [_z / x]_B -> A.x w = [_z / x]_B)
24	20, 23	hban 1356	. . . . . 6 |- ((z e. A /\ w = [_z / x]_B) -> A.x(z e. A /\ w = [_z / x]_B))
25		eleq1 1957	. . . . . . 7 |- (x = z -> (x e. A <-> z e. A))
26		csbeq1a 2546	. . . . . . . 8 |- (x = z -> B = [_z / x]_B)
27	26	eqeq2d 1895	. . . . . . 7 |- (x = z -> (w = B <-> w = [_z / x]_B))
28	25, 27	anbi12d 690	. . . . . 6 |- (x = z -> ((x e. A /\ w = B) <-> (z e. A /\ w = [_z / x]_B)))
29	24, 28	equsex 1513	. . . . 5 |- (E.x(x = z /\ (x e. A /\ w = B)) <-> (z e. A /\ w = [_z / x]_B))
30	17, 19, 29	3bitr2ri 197	. . . 4 |- ((z e. A /\ w = [_z / x]_B) <-> E.x e. A (x = z /\ w = B))
31	16, 30	bitr4i 193	. . 3 |- (E.x e. A <.z, w>. e. {<.x, B>.} <-> (z e. A /\ w = [_z / x]_B))
32		eliun 3259	. . 3 |- (<.z, w>. e. U_x e. A {<.x, B>.} <-> E.x e. A <.z, w>. e. {<.x, B>.})
33		ax-17 1317	. . . . . . 7 |- (y e. z -> A.x y e. z)
34	7, 33	hbcsb1 2568	. . . . . 6 |- (y e. [_z / x]_B -> A.x y e. [_z / x]_B)
35	34	hbeleq 1997	. . . . 5 |- (y = [_z / x]_B -> A.x y = [_z / x]_B)
36	20, 35	hban 1356	. . . 4 |- ((z e. A /\ y = [_z / x]_B) -> A.x(z e. A /\ y = [_z / x]_B))
37		ax-17 1317	. . . . 5 |- (z e. A -> A.y z e. A)
38		ax-17 1317	. . . . 5 |- (w = [_z / x]_B -> A.y w = [_z / x]_B)
39	37, 38	hban 1356	. . . 4 |- ((z e. A /\ w = [_z / x]_B) -> A.y(z e. A /\ w = [_z / x]_B))
40	26	eqeq2d 1895	. . . . 5 |- (x = z -> (y = B <-> y = [_z / x]_B))
41	25, 40	anbi12d 690	. . . 4 |- (x = z -> ((x e. A /\ y = B) <-> (z e. A /\ y = [_z / x]_B)))
42		eqeq1 1890	. . . . 5 |- (y = w -> (y = [_z / x]_B <-> w = [_z / x]_B))
43	42	anbi2d 678	. . . 4 |- (y = w -> ((z e. A /\ y = [_z / x]_B) <-> (z e. A /\ w = [_z / x]_B)))
44	36, 39, 7, 8, 41, 43	opelopabf 3572	. . 3 |- (<.z, w>. e. {<.x, y>. | (x e. A /\ y = B)} <-> (z e. A /\ w = [_z / x]_B))
45	31, 32, 44	3bitr4i 200	. 2 |- (<.z, w>. e. U_x e. A {<.x, B>.} <-> <.z, w>. e. {<.x, y>. | (x e. A /\ y = B)})
46	5, 6, 45	eqrelriv 4080	1 |- U_x e. A {<.x, B>.} = {<.x, y>. | (x e. A /\ y = B)}

Syntax hints:   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  E.wrex 2106  _Vcvv 2292  [_csb 2540  {csn 3044  <.cop 3046  U_ciun 3255  {copab 3395  Rel wrel 3991

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-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  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-iun 3257  df-opab 3396  df-xp 4000  df-rel 4001

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