Theorem iunfoprab 5072

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

Hypothesis

Ref	Expression
iunfoprab.1	|- C e. _V

Assertion

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

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

Proof of Theorem iunfoprab

Step	Hyp	Ref	Expression
1		fvex 4689	. . . 4 |- (1st` w) e. _V
2		ax-17 1317	. . . 4 |- (v e. (1st` w) -> A.x v e. (1st` w))
3	1, 2	hbcsb1 2568	. . 3 |- (v e. [_(1st` w) / x]_[_(2nd` w) / y]_C -> A.x v e. [_(1st` w) / x]_[_(2nd` w) / y]_C)
4		ax-17 1317	. . . . 5 |- (v e. (1st` w) -> A.y v e. (1st` w))
5		fvex 4689	. . . . . 6 |- (2nd` w) e. _V
6		ax-17 1317	. . . . . 6 |- (v e. (2nd` w) -> A.y v e. (2nd` w))
7	5, 6	hbcsb1 2568	. . . . 5 |- (v e. [_(2nd` w) / y]_C -> A.y v e. [_(2nd` w) / y]_C)
8	4, 7	hbcsbg 2569	. . . 4 |- ((1st` w) e. _V -> (v e. [_(1st` w) / x]_[_(2nd` w) / y]_C -> A.y v e. [_(1st` w) / x]_[_(2nd` w) / y]_C))
9	1, 8	ax-mp 7	. . 3 |- (v e. [_(1st` w) / x]_[_(2nd` w) / y]_C -> A.y v e. [_(1st` w) / x]_[_(2nd` w) / y]_C)
10		csbeq1a 2546	. . . 4 |- (y = (2nd` w) -> C = [_(2nd` w) / y]_C)
11		csbeq1a 2546	. . . 4 |- (x = (1st` w) -> [_(2nd` w) / y]_C = [_(1st` w) / x]_[_(2nd` w) / y]_C)
12	10, 11	sylan9eqr 1951	. . 3 |- ((x = (1st` w) /\ y = (2nd` w)) -> C = [_(1st` w) / x]_[_(2nd` w) / y]_C)
13	3, 9, 12	dfoprab5sf 5058	. 2 |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (A X. B) /\ z = [_(1st` w) / x]_[_(2nd` w) / y]_C)}
14		iunfoprab.1	. . . . 5 |- C e. _V
15	5, 14	csbex 2549	. . . 4 |- [_(2nd` w) / y]_C e. _V
16	1, 15	csbex 2549	. . 3 |- [_(1st` w) / x]_[_(2nd` w) / y]_C e. _V
17	16	iunfopab 4542	. 2 |- U_w e. (A X. B){<.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.} = {<.w, z>. | (w e. (A X. B) /\ z = [_(1st` w) / x]_[_(2nd` w) / y]_C)}
18		ax-17 1317	. . . . 5 |- (v e. w -> A.x v e. w)
19	18, 3	hbop 3168	. . . 4 |- (v e. <.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>. -> A.x v e. <.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.)
20	19	hbsn 3088	. . 3 |- (v e. {<.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.} -> A.x v e. {<.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.})
21		ax-17 1317	. . . . 5 |- (v e. w -> A.y v e. w)
22	21, 9	hbop 3168	. . . 4 |- (v e. <.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>. -> A.y v e. <.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.)
23	22	hbsn 3088	. . 3 |- (v e. {<.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.} -> A.y v e. {<.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.})
24		ax-17 1317	. . 3 |- (v e. {<.<.x, y>., C>.} -> A.w v e. {<.<.x, y>., C>.})
25		id 73	. . . . 5 |- (w = <.x, y>. -> w = <.x, y>.)
26		csbopeq1a 5052	. . . . . . 7 |- (<.x, y>. = w -> C = [_(1st` w) / x]_[_(2nd` w) / y]_C)
27	26	eqcoms 1887	. . . . . 6 |- (w = <.x, y>. -> C = [_(1st` w) / x]_[_(2nd` w) / y]_C)
28	27	eqcomd 1889	. . . . 5 |- (w = <.x, y>. -> [_(1st` w) / x]_[_(2nd` w) / y]_C = C)
29	25, 28	opeq12d 3166	. . . 4 |- (w = <.x, y>. -> <.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>. = <.<.x, y>., C>.)
30	29	sneqd 3056	. . 3 |- (w = <.x, y>. -> {<.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.} = {<.<.x, y>., C>.})
31	20, 23, 24, 30	iunxpf 4045	. 2 |- U_w e. (A X. B){<.w, [_(1st` w) / x]_[_(2nd` w) / y]_C>.} = U_x e. A U_y e. B {<.<.x, y>., C>.}
32	13, 17, 31	3eqtr2ri 1916	1 |- U_x e. A U_y e. B {<.<.x, y>., C>.} = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}

Syntax hints:   -> wi 3   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  _Vcvv 2292  [_csb 2540  {csn 3044  <.cop 3046  U_ciun 3255  {copab 3395   X. cxp 3984  ` cfv 3998  {copab2 4885  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  dfmpt2 5074  fpar 5085

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-13 1311  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  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  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-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-oprab 4887  df-1st 5020  df-2nd 5021

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