Theorem ofeqd 6268

Description: Equality theorem for function operation, deduction form. (Contributed by SN, 11-Nov-2024.)

Hypothesis

Ref	Expression
ofeqd.1	|- ( ph -> R = S )

Assertion

Ref	Expression
ofeqd	|- ( ph -> oF R = oF S )

Proof of Theorem ofeqd

Dummy variables f g x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofeqd.1	. . . . 5 |- ( ph -> R = S )
2	1	oveqd 6067	. . . 4 |- ( ph -> ( ( f ` x ) R ( g ` x ) ) = ( ( f ` x ) S ( g ` x ) ) )
3	2	mpteq2dv 4201	. . 3 |- ( ph -> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) = ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) S ( g ` x ) ) ) )
4	3	mpoeq3dv 6119	. 2 |- ( ph -> ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) ) = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) S ( g ` x ) ) ) ) )
5		df-of 6266	. 2 |- oF R = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
6		df-of 6266	. 2 |- oF S = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) S ( g ` x ) ) ) )
7	4, 5, 6	3eqtr4g 2290	1 |- ( ph -> oF R = oF S )

Syntax hints:   -> wi 4   = wceq 1398   _Vcvv 2813   i^i cin 3210   |-> cmpt 4171   dom cdm 4749   ` cfv 5352  (class class class)co 6050   e. cmpo 6052   oFcof 6264

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-iota 5312  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-of 6266

This theorem is referenced by:  psrval  14814  lgseisenlem3  15945  lgseisenlem4  15946