Theorem mpt2eq123i 6718

Description: An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)

Hypotheses

Ref	Expression
mpt2eq123i.1	|- A = D
mpt2eq123i.2	|- B = E
mpt2eq123i.3	|- C = F

Assertion

Ref	Expression
mpt2eq123i	|- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F )

Proof of Theorem mpt2eq123i

Step	Hyp	Ref	Expression
1		mpt2eq123i.1	. . . 4 |- A = D
2	1	a1i 11	. . 3 |- ( T. -> A = D )
3		mpt2eq123i.2	. . . 4 |- B = E
4	3	a1i 11	. . 3 |- ( T. -> B = E )
5		mpt2eq123i.3	. . . 4 |- C = F
6	5	a1i 11	. . 3 |- ( T. -> C = F )
7	2, 4, 6	mpt2eq123dv 6717	. 2 |- ( T. -> ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F ) )
8	7	trud 1493	1 |- ( x e. A , y e. B |-> C ) = ( x e. D , y e. E |-> F )

Syntax hints:   = wceq 1483   T. wtru 1484   |-> 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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  ofmres  7164  seqval  12812  oppgtmd  21901  wlkson  26552  mdetlap1  29892  sdc  33540  tgrpset  36033  mendvscafval  37760  fsovcnvlem  38307  hspmbl  40843