Theorem mpteq1i 4218

Description: An equality theorem for the maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) Remove all disjoint variable conditions. (Revised by SN, 11-Nov-2024.)

Hypothesis

Ref	Expression
mpteq1i.1	|- A = B

Assertion

Ref	Expression
mpteq1i	|- ( x e. A |-> C ) = ( x e. B |-> C )

Proof of Theorem mpteq1i

Step	Hyp	Ref	Expression
1		mpteq1i.1	. . . 4 |- A = B
2	1	a1i 9	. . 3 |- ( T. -> A = B )
3		eqidd 2239	. . 3 |- ( T. -> C = C )
4	2, 3	mpteq12dv 4214	. 2 |- ( T. -> ( x e. A |-> C ) = ( x e. B |-> C ) )
5	4	mptru 1411	1 |- ( x e. A |-> C ) = ( x e. B |-> C )

This proof depends on syntax axioms:   = wceq 1402   T. wtru 1403   |-> cmpt 4193

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-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-ral 2533  df-opab 4194  df-mpt 4195

This theorem is used by:  gsum2dfilem2  14272  gsumcom2fi  14275