Theorem dfoprab4s 5056

Description: Operation class abstraction expressed without existential quantifiers.

Hypothesis

Ref	Expression
dfoprab4s.1	|- (<.x, y>. = w -> C = D)

Assertion

Ref	Expression
dfoprab4s	|- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (A X. B) /\ z = D)}

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

Proof of Theorem dfoprab4s

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (<.x, y>. = w -> (<.x, y>. e. (A X. B) <-> w e. (A X. B)))
2		visset 2295	. . . . . 6 |- y e. _V
3	2	opelxp 4036	. . . . 5 |- (<.x, y>. e. (A X. B) <-> (x e. A /\ y e. B))
4	1, 3	syl5bbr 593	. . . 4 |- (<.x, y>. = w -> ((x e. A /\ y e. B) <-> w e. (A X. B)))
5		dfoprab4s.1	. . . . 5 |- (<.x, y>. = w -> C = D)
6	5	eqeq2d 1895	. . . 4 |- (<.x, y>. = w -> (z = C <-> z = D))
7	4, 6	anbi12d 690	. . 3 |- (<.x, y>. = w -> (((x e. A /\ y e. B) /\ z = C) <-> (w e. (A X. B) /\ z = D)))
8	7	dfoprab3s 5055	. 2 |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (_V X. _V) /\ (w e. (A X. B) /\ z = D))}
9		xpss 4056	. . . . . 6 |- (A X. B) C_ (_V X. _V)
10	9	sseli 2617	. . . . 5 |- (w e. (A X. B) -> w e. (_V X. _V))
11	10	adantr 425	. . . 4 |- ((w e. (A X. B) /\ z = D) -> w e. (_V X. _V))
12	11	pm4.71ri 700	. . 3 |- ((w e. (A X. B) /\ z = D) <-> (w e. (_V X. _V) /\ (w e. (A X. B) /\ z = D)))
13	12	opabbii 3402	. 2 |- {<.w, z>. | (w e. (A X. B) /\ z = D)} = {<.w, z>. | (w e. (_V X. _V) /\ (w e. (A X. B) /\ z = D))}
14	8, 13	eqtr4i 1911	1 |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.w, z>. | (w e. (A X. B) /\ z = D)}

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292  <.cop 3046  {copab 3395   X. cxp 3984  {copab2 4885

This theorem is referenced by:  dfoprab5s 5057  txcnoprab 15911

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-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-oprab 4887  df-1st 5020  df-2nd 5021

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