Theorem ofeqd 6294

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 6092	. . . 4 |- ( ph -> ( ( f ` x ) R ( g ` x ) ) = ( ( f ` x ) S ( g ` x ) ) )
3	2	mpteq2dv 4217	. . 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 6144	. 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 6292	. 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 6292	. 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 2296	1 |- ( ph -> oF R = oF S )

Syntax hints:   -> wi 4   = wceq 1402   _Vcvv 2821   i^i cin 3219   |-> cmpt 4187   dom cdm 4769   ` cfv 5372  (class class class)co 6075   e. cmpo 6077   oFcof 6290

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem 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-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-iota 5332  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292

This theorem is referenced by:  psrval  14973  lgseisenlem3  16105  lgseisenlem4  16106