Theorem dfoprab4 4122

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

Hypothesis

Ref	Expression
dfoprab4.1	|- ((x = (1st` w) /\ y = (2nd` w)) -> C = D)

Assertion

Ref	Expression
dfoprab4	|- {<.<.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 dfoprab4

Step	Hyp	Ref	Expression
1		dfoprab4.1	. . 3 |- ((x = (1st` w) /\ y = (2nd` w)) -> C = D)
2		fveq2 3730	. . . 4 |- (<.x, y>. = w -> (1st` <.x, y>.) = (1st` w))
3		visset 1816	. . . . 5 |- x e. V
4	3	op1st 4091	. . . 4 |- (1st` <.x, y>.) = x
5	2, 4	syl5eqr 1524	. . 3 |- (<.x, y>. = w -> x = (1st` w))
6		fveq2 3730	. . . 4 |- (<.x, y>. = w -> (2nd` <.x, y>.) = (2nd` w))
7		visset 1816	. . . . 5 |- y e. V
8	3, 7	op2nd 4092	. . . 4 |- (2nd` <.x, y>.) = y
9	6, 8	syl5eqr 1524	. . 3 |- (<.x, y>. = w -> y = (2nd` w))
10	1, 5, 9	sylanc 473	. 2 |- (<.x, y>. = w -> C = D)
11	10	dfoprab5 4121	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 223   = wceq 958   e. wcel 960  <.cop 2415  {copab 2671   X. cxp 3174  ` cfv 3188  {copab2 3970  1stc1st 4083  2ndc2nd 4084

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-oprab 3972  df-1st 4085  df-2nd 4086

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