Theorem fconstopab 3210

Description: Representation of a constant function using ordered pairs.

Assertion

Ref	Expression
fconstopab	|- (A X. {B}) = {<.x, y>. | (x e. A /\ y = B)}

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem fconstopab

Step	Hyp	Ref	Expression
1		df-xp 3184	. 2 |- (A X. {B}) = {<.x, y>. | (x e. A /\ y e. {B})}
2		elsn 2421	. . . 4 |- (y e. {B} <-> y = B)
3	2	anbi2i 480	. . 3 |- ((x e. A /\ y e. {B}) <-> (x e. A /\ y = B))
4	3	opabbii 2671	. 2 |- {<.x, y>. | (x e. A /\ y e. {B})} = {<.x, y>. | (x e. A /\ y = B)}
5	1, 4	eqtr 1495	1 |- (A X. {B}) = {<.x, y>. | (x e. A /\ y = B)}

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958  {csn 2409  {copab 2666   X. cxp 3168

This theorem is referenced by:  serzclim0 7109  efcltlem2 7305  occllem5 9177  bra0 9874

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-10 966  ax-12 968  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

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-sn 2412  df-opab 2667  df-xp 3184

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