Theorem dfoprab4 4116

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 3724	. . . 4 |- (<.x, y>. = w -> (1st` <.x, y>.) = (1st` w))
3		visset 1813	. . . . 5 |- x e. V
4	3	op1st 4085	. . . 4 |- (1st` <.x, y>.) = x
5	2, 4	syl5eqr 1521	. . 3 |- (<.x, y>. = w -> x = (1st` w))
6		fveq2 3724	. . . 4 |- (<.x, y>. = w -> (2nd` <.x, y>.) = (2nd` w))
7		visset 1813	. . . . 5 |- y e. V
8	3, 7	op2nd 4086	. . . 4 |- (2nd` <.x, y>.) = y
9	6, 8	syl5eqr 1521	. . 3 |- (<.x, y>. = w -> y = (2nd` w))
10	1, 5, 9	sylanc 471	. 2 |- (<.x, y>. = w -> C = D)
11	10	dfoprab5 4115	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 956   e. wcel 958  <.cop 2411  {copab 2666   X. cxp 3168  ` cfv 3182  {copab2 3964  1stc1st 4077  2ndc2nd 4078

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-3an 777  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-csb 2002  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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