Theorem dfoprab3 4114

Description: A way to define an operation class abstraction without using existential quantifiers.

Assertion

Ref	Expression
dfoprab3	|- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)}

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

Proof of Theorem dfoprab3

Step	Hyp	Ref	Expression
1		dfoprab2 3991	. 2 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}
2		fvex 3732	. . . . . 6 |- (1st` w) e. V
3	2	hbsbc1v 1950	. . . . 5 |- ([(1st` w) / x][(2nd` w) / y]ph -> A.x[(1st` w) / x][(2nd` w) / y]ph)
4	3	19.41 1095	. . . 4 |- (E.x(E.y w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph) <-> (E.xE.y w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
5		fveq2 3724	. . . . . . . . . . 11 |- (w = <.x, y>. -> (2nd` w) = (2nd` <.x, y>.))
6		visset 1813	. . . . . . . . . . . 12 |- x e. V
7		visset 1813	. . . . . . . . . . . 12 |- y e. V
8	6, 7	op2nd 4086	. . . . . . . . . . 11 |- (2nd` <.x, y>.) = y
9	5, 8	syl6req 1524	. . . . . . . . . 10 |- (w = <.x, y>. -> y = (2nd` w))
10		sbceq1a 1944	. . . . . . . . . 10 |- (y = (2nd` w) -> (ph <-> [(2nd` w) / y]ph))
11	9, 10	syl 10	. . . . . . . . 9 |- (w = <.x, y>. -> (ph <-> [(2nd` w) / y]ph))
12		fveq2 3724	. . . . . . . . . . 11 |- (w = <.x, y>. -> (1st` w) = (1st` <.x, y>.))
13	6	op1st 4085	. . . . . . . . . . 11 |- (1st` <.x, y>.) = x
14	12, 13	syl6req 1524	. . . . . . . . . 10 |- (w = <.x, y>. -> x = (1st` w))
15		sbceq1a 1944	. . . . . . . . . 10 |- (x = (1st` w) -> ([(2nd` w) / y]ph <-> [(1st` w) / x][(2nd` w) / y]ph))
16	14, 15	syl 10	. . . . . . . . 9 |- (w = <.x, y>. -> ([(2nd` w) / y]ph <-> [(1st` w) / x][(2nd` w) / y]ph))
17	11, 16	bitrd 528	. . . . . . . 8 |- (w = <.x, y>. -> (ph <-> [(1st` w) / x][(2nd` w) / y]ph))
18	17	pm5.32i 645	. . . . . . 7 |- ((w = <.x, y>. /\ ph) <-> (w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
19	18	exbii 1051	. . . . . 6 |- (E.y(w = <.x, y>. /\ ph) <-> E.y(w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
20		ax-17 971	. . . . . . . . 9 |- (x e. (1st` w) -> A.y x e. (1st` w))
21		fvex 3732	. . . . . . . . . 10 |- (2nd` w) e. V
22	21	hbsbc1v 1950	. . . . . . . . 9 |- ([(2nd` w) / y]ph -> A.y[(2nd` w) / y]ph)
23	20, 22	hbsbcg 1951	. . . . . . . 8 |- ((1st` w) e. V -> ([(1st` w) / x][(2nd` w) / y]ph -> A.y[(1st` w) / x][(2nd` w) / y]ph))
24	2, 23	ax-mp 7	. . . . . . 7 |- ([(1st` w) / x][(2nd` w) / y]ph -> A.y[(1st` w) / x][(2nd` w) / y]ph)
25	24	19.41 1095	. . . . . 6 |- (E.y(w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph) <-> (E.y w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
26	19, 25	bitr 173	. . . . 5 |- (E.y(w = <.x, y>. /\ ph) <-> (E.y w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
27	26	exbii 1051	. . . 4 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.x(E.y w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
28		elvv 3228	. . . . 5 |- (w e. (V X. V) <-> E.xE.y w = <.x, y>.)
29	28	anbi1i 481	. . . 4 |- ((w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph) <-> (E.xE.y w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
30	4, 27, 29	3bitr4 183	. . 3 |- (E.xE.y(w = <.x, y>. /\ ph) <-> (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph))
31	30	opabbii 2671	. 2 |- {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)} = {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)}
32	1, 31	eqtr 1495	1 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  [wsbc 1170  Vcvv 1811  <.cop 2411  {copab 2666   X. cxp 3168  ` cfv 3182  {copab2 3964  1stc1st 4077  2ndc2nd 4078

This theorem is referenced by:  dfoprab5 4115  eloprabi 4118  foprab2 4119

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-oprab 3966  df-1st 4079  df-2nd 4080

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