Theorem fconstmpo 6148

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 4797	. 2 |- ( ( A X. B ) X. { C } ) = ( z e. ( A X. B ) |-> C )
2		eqidd 2233	. . 3 |- ( z = <. x , y >. -> C = C )
3	2	mpompt 6145	. 2 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> C )
4	1, 3	eqtri 2253	1 |- ( ( A X. B ) X. { C } ) = ( x e. A , y e. B |-> C )

Syntax hints:   = wceq 1398   {csn 3689   <.cop 3692   |-> cmpt 4171   X. cxp 4747   e. cmpo 6052

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-iun 3993  df-opab 4172  df-mpt 4173  df-xp 4755  df-rel 4756  df-oprab 6054  df-mpo 6055

This theorem is referenced by: (None)