Theorem fconstmpo 6098

Description: Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)

Assertion

Ref	Expression
fconstmpo	|- ( ( A X. B ) X. { C } ) = ( x e. A , y e. B |-> C )

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

Proof of Theorem fconstmpo

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstmpt 4765	. 2 |- ( ( A X. B ) X. { C } ) = ( z e. ( A X. B ) |-> C )
2		eqidd 2230	. . 3 |- ( z = <. x , y >. -> C = C )
3	2	mpompt 6095	. 2 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> C )
4	1, 3	eqtri 2250	1 |- ( ( A X. B ) X. { C } ) = ( x e. A , y e. B |-> C )

Syntax hints:   = wceq 1395   {csn 3666   <.cop 3669   |-> cmpt 4144   X. cxp 4716   e. cmpo 6002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-iun 3966  df-opab 4145  df-mpt 4146  df-xp 4724  df-rel 4725  df-oprab 6004  df-mpo 6005

This theorem is referenced by: (None)