Theorem dfoprab3 4107

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 3986	. 2 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}
2		fvex 3727	. . . . . 6 |- (1st` w) e. V
3	2	hbsbc1v 1947	. . . . 5 |- ([(1st` w) / x][(2nd` w) / y]ph -> A.x[(1st` w) / x][(2nd` w) / y]ph)
4	3	19.41 1094	. . . 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 3719	. . . . . . . . . . 11 |- (w = <.x, y>. -> (2nd` w) = (2nd` <.x, y>.))
6		visset 1810	. . . . . . . . . . . 12 |- x e. V
7		visset 1810	. . . . . . . . . . . 12 |- y e. V
8	6, 7	op2nd 4079	. . . . . . . . . . 11 |- (2nd` <.x, y>.) = y
9	5, 8	syl6req 1522	. . . . . . . . . 10 |- (w = <.x, y>. -> y = (2nd` w))
10		sbceq1a 1941	. . . . . . . . . 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 3719	. . . . . . . . . . 11 |- (w = <.x, y>. -> (1st` w) = (1st` <.x, y>.))
13	6	op1st 4078	. . . . . . . . . . 11 |- (1st` <.x, y>.) = x
14	12, 13	syl6req 1522	. . . . . . . . . 10 |- (w = <.x, y>. -> x = (1st` w))
15		sbceq1a 1941	. . . . . . . . . 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 527	. . . . . . . 8 |- (w = <.x, y>. -> (ph <-> [(1st` w) / x][(2nd` w) / y]ph))
18	17	pm5.32i 644	. . . . . . 7 |- ((w = <.x, y>. /\ ph) <-> (w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
19	18	exbii 1050	. . . . . 6 |- (E.y(w = <.x, y>. /\ ph) <-> E.y(w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
20		ax-17 970	. . . . . . . . 9 |- (x e. (1st` w) -> A.y x e. (1st` w))
21		fvex 3727	. . . . . . . . . 10 |- (2nd` w) e. V
22	21	hbsbc1v 1947	. . . . . . . . 9 |- ([(2nd` w) / y]ph -> A.y[(2nd` w) / y]ph)
23	20, 22	hbsbcg 1948	. . . . . . . 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 1094	. . . . . 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 1050	. . . 4 |- (E.xE.y(w = <.x, y>. /\ ph) <-> E.x(E.y w = <.x, y>. /\ [(1st` w) / x][(2nd` w) / y]ph))
28		elvv 3224	. . . . 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 2667	. 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 1493	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 953   = wceq 955   e. wcel 957  E.wex 979  [wsbc 1169  Vcvv 1808  <.cop 2408  {copab 2662   X. cxp 3164  ` cfv 3178  {copab2 3959  1stc1st 4070  2ndc2nd 4071

This theorem is referenced by:  dfoprab5 4108  eloprabi 4111  foprab2 4112

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-sbc 1939  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fv 3194  df-oprab 3961  df-1st 4072  df-2nd 4073

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