Description: Lemma for showing the equality of values for functions like slot extractors E at a proper class. Extracted from several former proofs of lemmas like zlmlem . (Contributed by AV, 31-Oct-2024)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | fveqprc.e | |- ( E ` (/) ) = (/) |
|
| fveqprc.y | |- Y = ( F ` X ) |
||
| Assertion | fveqprc | |- ( -. X e. _V -> ( E ` X ) = ( E ` Y ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveqprc.e | |- ( E ` (/) ) = (/) |
|
| 2 | fveqprc.y | |- Y = ( F ` X ) |
|
| 3 | 1 | eqcomi | |- (/) = ( E ` (/) ) |
| 4 | fvprc | |- ( -. X e. _V -> ( E ` X ) = (/) ) |
|
| 5 | fvprc | |- ( -. X e. _V -> ( F ` X ) = (/) ) |
|
| 6 | 2 5 | eqtrid | |- ( -. X e. _V -> Y = (/) ) |
| 7 | 6 | fveq2d | |- ( -. X e. _V -> ( E ` Y ) = ( E ` (/) ) ) |
| 8 | 3 4 7 | 3eqtr4a | |- ( -. X e. _V -> ( E ` X ) = ( E ` Y ) ) |