Theorem fconstmpo 6183

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 4822	. 2 |- ( ( A X. B ) X. { C } ) = ( z e. ( A X. B ) |-> C )
2		eqidd 2239	. . 3 |- ( z = <. x , y >. -> C = C )
3	2	mpompt 6180	. 2 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> C )
4	1, 3	eqtri 2259	1 |- ( ( A X. B ) X. { C } ) = ( x e. A , y e. B |-> C )

This proof depends on syntax axioms:   = wceq 1402   {csn 3709   <.cop 3712   |-> cmpt 4192   X. cxp 4772   e. cmpo 6087

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-iun 4014  df-opab 4193  df-mpt 4194  df-xp 4780  df-rel 4781  df-oprab 6089  df-mpo 6090

This theorem is used by: (None)