Theorem ofeqd 6183

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 5984	. . . 4 |- ( ph -> ( ( f ` x ) R ( g ` x ) ) = ( ( f ` x ) S ( g ` x ) ) )
3	2	mpteq2dv 4151	. . 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 6034	. 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 6181	. 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 6181	. 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 2265	1 |- ( ph -> oF R = oF S )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2776   i^i cin 3173   |-> cmpt 4121   dom cdm 4693   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   oFcof 6179

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-uni 3865  df-br 4060  df-opab 4122  df-mpt 4123  df-iota 5251  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181

This theorem is referenced by:  psrval  14543  lgseisenlem3  15664  lgseisenlem4  15665