Theorem mpoeq3dv 5954

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 1019	. 2 |- ( ( ph /\ x e. A /\ y e. B ) -> C = D )
3	2	mpoeq3dva 5952	1 |- ( ph -> ( x e. A , y e. B |-> C ) = ( x e. A , y e. B |-> D ) )

Syntax hints:   -> wi 4   = wceq 1363   e. wcel 2158   e. cmpo 5890

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-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-11 1516  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-oprab 5892  df-mpo 5893

This theorem is referenced by:  prdsex  12735