Theorem dfopab2 5053

Description: A way to define an ordered-pair class abstraction without using existential quantifiers.

Assertion

Ref	Expression
dfopab2	|- {<.x, y>. | ph} = {z | (z e. (_V X. _V) /\ [(1st` z) / x][(2nd` z) / y]ph)}

Distinct variable groups:   ph,z   x,y,z

Proof of Theorem dfopab2

Step	Hyp	Ref	Expression
1		df-opab 3396	. 2 |- {<.x, y>. | ph} = {z | E.xE.y(z = <.x, y>. /\ ph)}
2		fvex 4689	. . . . . 6 |- (1st` z) e. _V
3	2	hbsbc1v 2464	. . . . 5 |- ([(1st` z) / x][(2nd` z) / y]ph -> A.x[(1st` z) / x][(2nd` z) / y]ph)
4	3	19.41 1448	. . . 4 |- (E.x(E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph) <-> (E.xE.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
5		fveq2 4681	. . . . . . . . . . 11 |- (z = <.x, y>. -> (2nd` z) = (2nd` <.x, y>.))
6		visset 2295	. . . . . . . . . . . 12 |- x e. _V
7		visset 2295	. . . . . . . . . . . 12 |- y e. _V
8	6, 7	op2nd 5027	. . . . . . . . . . 11 |- (2nd` <.x, y>.) = y
9	5, 8	syl6req 1945	. . . . . . . . . 10 |- (z = <.x, y>. -> y = (2nd` z))
10		sbceq1a 2456	. . . . . . . . . 10 |- (y = (2nd` z) -> (ph <-> [(2nd` z) / y]ph))
11	9, 10	syl 12	. . . . . . . . 9 |- (z = <.x, y>. -> (ph <-> [(2nd` z) / y]ph))
12		fveq2 4681	. . . . . . . . . . 11 |- (z = <.x, y>. -> (1st` z) = (1st` <.x, y>.))
13	6	op1st 5026	. . . . . . . . . . 11 |- (1st` <.x, y>.) = x
14	12, 13	syl6req 1945	. . . . . . . . . 10 |- (z = <.x, y>. -> x = (1st` z))
15		sbceq1a 2456	. . . . . . . . . 10 |- (x = (1st` z) -> ([(2nd` z) / y]ph <-> [(1st` z) / x][(2nd` z) / y]ph))
16	14, 15	syl 12	. . . . . . . . 9 |- (z = <.x, y>. -> ([(2nd` z) / y]ph <-> [(1st` z) / x][(2nd` z) / y]ph))
17	11, 16	bitrd 587	. . . . . . . 8 |- (z = <.x, y>. -> (ph <-> [(1st` z) / x][(2nd` z) / y]ph))
18	17	pm5.32i 707	. . . . . . 7 |- ((z = <.x, y>. /\ ph) <-> (z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
19	18	exbii 1398	. . . . . 6 |- (E.y(z = <.x, y>. /\ ph) <-> E.y(z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
20		ax-17 1317	. . . . . . . . 9 |- (x e. (1st` z) -> A.y x e. (1st` z))
21		fvex 4689	. . . . . . . . . 10 |- (2nd` z) e. _V
22	21	hbsbc1v 2464	. . . . . . . . 9 |- ([(2nd` z) / y]ph -> A.y[(2nd` z) / y]ph)
23	20, 22	hbsbcg 2466	. . . . . . . 8 |- ((1st` z) e. _V -> ([(1st` z) / x][(2nd` z) / y]ph -> A.y[(1st` z) / x][(2nd` z) / y]ph))
24	2, 23	ax-mp 7	. . . . . . 7 |- ([(1st` z) / x][(2nd` z) / y]ph -> A.y[(1st` z) / x][(2nd` z) / y]ph)
25	24	19.41 1448	. . . . . 6 |- (E.y(z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph) <-> (E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
26	19, 25	bitri 190	. . . . 5 |- (E.y(z = <.x, y>. /\ ph) <-> (E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
27	26	exbii 1398	. . . 4 |- (E.xE.y(z = <.x, y>. /\ ph) <-> E.x(E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
28		elvv 4053	. . . . 5 |- (z e. (_V X. _V) <-> E.xE.y z = <.x, y>.)
29	28	anbi1i 539	. . . 4 |- ((z e. (_V X. _V) /\ [(1st` z) / x][(2nd` z) / y]ph) <-> (E.xE.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
30	4, 27, 29	3bitr4i 200	. . 3 |- (E.xE.y(z = <.x, y>. /\ ph) <-> (z e. (_V X. _V) /\ [(1st` z) / x][(2nd` z) / y]ph))
31	30	abbii 2006	. 2 |- {z | E.xE.y(z = <.x, y>. /\ ph)} = {z | (z e. (_V X. _V) /\ [(1st` z) / x][(2nd` z) / y]ph)}
32	1, 31	eqtri 1908	1 |- {<.x, y>. | ph} = {z | (z e. (_V X. _V) /\ [(1st` z) / x][(2nd` z) / y]ph)}

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  {cab 1871  _Vcvv 2292  <.cop 3046  {copab 3395   X. cxp 3984  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  fparlem1 5081  fparlem2 5082

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-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-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-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-1st 5020  df-2nd 5021

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