Theorem xpsnopab 41765

Description: A Cartesian product with a singleton expressed as ordered-pair class abstraction. (Contributed by AV, 27-Jan-2020.)

Assertion

Ref	Expression
xpsnopab	|- ( { X } X. C ) = { <. a , b >. | ( a = X /\ b e. C ) }

Distinct variable groups:   C, a, b   X, a, b

Proof of Theorem xpsnopab

Step	Hyp	Ref	Expression
1		df-xp 5120	. 2 |- ( { X } X. C ) = { <. a , b >. | ( a e. { X } /\ b e. C ) }
2		velsn 4193	. . . 4 |- ( a e. { X } <-> a = X )
3	2	anbi1i 731	. . 3 |- ( ( a e. { X } /\ b e. C ) <-> ( a = X /\ b e. C ) )
4	3	opabbii 4717	. 2 |- { <. a , b >. | ( a e. { X } /\ b e. C ) } = { <. a , b >. | ( a = X /\ b e. C ) }
5	1, 4	eqtri 2644	1 |- ( { X } X. C ) = { <. a , b >. | ( a = X /\ b e. C ) }

Syntax hints:   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   {copab 4712   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178  df-opab 4713  df-xp 5120

This theorem is referenced by:  xpiun  41766