Theorem fconstmpt2 6755

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

Assertion

Ref	Expression
fconstmpt2	|- ( ( 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 fconstmpt2

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fconstmpt 5163	. 2 |- ( ( A X. B ) X. { C } ) = ( z e. ( A X. B ) |-> C )
2		eqidd 2623	. . 3 |- ( z = <. x , y >. -> C = C )
3	2	mpt2mpt 6752	. 2 |- ( z e. ( A X. B ) |-> C ) = ( x e. A , y e. B |-> C )
4	1, 3	eqtri 2644	1 |- ( ( A X. B ) X. { C } ) = ( x e. A , y e. B |-> C )

Syntax hints:   = wceq 1483   {csn 4177   <.cop 4183   |-> cmpt 4729   X. cxp 5112   |-> cmpt2 6652

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iun 4522  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  tposconst  7390  mat0op  20225  matsc  20256  mdetrsca2  20410  smadiadetglem2  20478