Theorem mpoeq3dv 6154

Description: An equality deduction for the maps-to notation restricted to the value of the operation. (Contributed by SO, 16-Jul-2018.)

Hypothesis

Ref	Expression
mpoeq3dv.1	|- ( ph -> C = D )

Assertion

Ref	Expression
mpoeq3dv	|- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Distinct variable groups:   ph, x   ph, y

Allowed substitution hints:   A( x, y)   B( x, y)   C( x, y)   D( x, y)

Proof of Theorem mpoeq3dv

Step	Hyp	Ref	Expression
1		mpoeq3dv.1	. . 3 |- ( ph -> C = D )
2	1	3ad2ant1 1049	. 2 |- ( ( ph /\ x e. A /\ y e. B ) -> C = D )
3	2	mpoeq3dva 6152	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   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-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

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-oprab 6089  df-mpo 6090

This theorem is used by:  ofeqd  6304  prdsex  14172