Theorem dfopab2 4103

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 2662	. 2 |- {<.x, y>. | ph} = {z | E.xE.y(z = <.x, y>. /\ ph)}
2		fvex 3723	. . . . . 6 |- (1st` z) e. V
3	2	hbsbc1v 1946	. . . . 5 |- ([(1st` z) / x][(2nd` z) / y]ph -> A.x[(1st` z) / x][(2nd` z) / y]ph)
4	3	19.41 1093	. . . 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 3715	. . . . . . . . . . 11 |- (z = <.x, y>. -> (2nd` z) = (2nd` <.x, y>.))
6		visset 1809	. . . . . . . . . . . 12 |- x e. V
7		visset 1809	. . . . . . . . . . . 12 |- y e. V
8	6, 7	op2nd 4076	. . . . . . . . . . 11 |- (2nd` <.x, y>.) = y
9	5, 8	syl6req 1521	. . . . . . . . . 10 |- (z = <.x, y>. -> y = (2nd` z))
10		sbceq1a 1940	. . . . . . . . . 10 |- (y = (2nd` z) -> (ph <-> [(2nd` z) / y]ph))
11	9, 10	syl 10	. . . . . . . . 9 |- (z = <.x, y>. -> (ph <-> [(2nd` z) / y]ph))
12		fveq2 3715	. . . . . . . . . . 11 |- (z = <.x, y>. -> (1st` z) = (1st` <.x, y>.))
13	6	op1st 4075	. . . . . . . . . . 11 |- (1st` <.x, y>.) = x
14	12, 13	syl6req 1521	. . . . . . . . . 10 |- (z = <.x, y>. -> x = (1st` z))
15		sbceq1a 1940	. . . . . . . . . 10 |- (x = (1st` z) -> ([(2nd` z) / y]ph <-> [(1st` z) / x][(2nd` z) / y]ph))
16	14, 15	syl 10	. . . . . . . . 9 |- (z = <.x, y>. -> ([(2nd` z) / y]ph <-> [(1st` z) / x][(2nd` z) / y]ph))
17	11, 16	bitrd 527	. . . . . . . 8 |- (z = <.x, y>. -> (ph <-> [(1st` z) / x][(2nd` z) / y]ph))
18	17	pm5.32i 644	. . . . . . 7 |- ((z = <.x, y>. /\ ph) <-> (z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
19	18	exbii 1049	. . . . . 6 |- (E.y(z = <.x, y>. /\ ph) <-> E.y(z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
20		ax-17 969	. . . . . . . . 9 |- (x e. (1st` z) -> A.y x e. (1st` z))
21		fvex 3723	. . . . . . . . . 10 |- (2nd` z) e. V
22	21	hbsbc1v 1946	. . . . . . . . 9 |- ([(2nd` z) / y]ph -> A.y[(2nd` z) / y]ph)
23	20, 22	hbsbcg 1947	. . . . . . . 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 1093	. . . . . 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	bitr 173	. . . . 5 |- (E.y(z = <.x, y>. /\ ph) <-> (E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
27	26	exbii 1049	. . . 4 |- (E.xE.y(z = <.x, y>. /\ ph) <-> E.x(E.y z = <.x, y>. /\ [(1st` z) / x][(2nd` z) / y]ph))
28		elvv 3223	. . . . 5 |- (z e. (V X. V) <-> E.xE.y z = <.x, y>.)
29	28	anbi1i 481	. . . 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	3bitr4 183	. . 3 |- (E.xE.y(z = <.x, y>. /\ ph) <-> (z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph))
31	30	abbii 1572	. 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	eqtr 1492	1 |- {<.x, y>. | ph} = {z | (z e. (V X. V) /\ [(1st` z) / x][(2nd` z) / y]ph)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  [wsbc 1168  {cab 1461  Vcvv 1807  <.cop 2407  {copab 2661   X. cxp 3163  ` cfv 3177  1stc1st 4067  2ndc2nd 4068

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-1st 4069  df-2nd 4070

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