Theorem fconstmpo 5966

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 4672	. 2 |- ( ( A X. B ) X. { C } ) = ( z e. ( A X. B ) |-> C )
2		eqidd 2178	. . 3 |- ( z = <. x , y >. -> C = C )
3	2	mpompt 5963	. 2 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> C )
4	1, 3	eqtri 2198	1 |- ( ( A X. B ) X. { C } ) = ( x e. A , y e. B |-> C )

Syntax hints:   = wceq 1353   {csn 3592   <.cop 3595   |-> cmpt 4063   X. cxp 4623   e. cmpo 5873

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-iun 3888  df-opab 4064  df-mpt 4065  df-xp 4631  df-rel 4632  df-oprab 5875  df-mpo 5876

This theorem is referenced by: (None)