# Metamath Proof Explorer

## Theorem etransclem12

Description: C applied to N . (Contributed by Glauco Siliprandi, 5-Apr-2020)

Ref Expression
Hypotheses etransclem12.1
`|- C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )`
etransclem12.2
`|- ( ph -> N e. NN0 )`
Assertion etransclem12
`|- ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )`

### Proof

Step Hyp Ref Expression
1 etransclem12.1
` |-  C = ( n e. NN0 |-> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } )`
2 etransclem12.2
` |-  ( ph -> N e. NN0 )`
3 oveq2
` |-  ( n = N -> ( 0 ... n ) = ( 0 ... N ) )`
4 3 oveq1d
` |-  ( n = N -> ( ( 0 ... n ) ^m ( 0 ... M ) ) = ( ( 0 ... N ) ^m ( 0 ... M ) ) )`
5 eqeq2
` |-  ( n = N -> ( sum_ j e. ( 0 ... M ) ( c ` j ) = n <-> sum_ j e. ( 0 ... M ) ( c ` j ) = N ) )`
6 4 5 rabeqbidv
` |-  ( n = N -> { c e. ( ( 0 ... n ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = n } = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )`
7 ovex
` |-  ( ( 0 ... N ) ^m ( 0 ... M ) ) e. _V`
8 7 rabex
` |-  { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } e. _V`
9 8 a1i
` |-  ( ph -> { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } e. _V )`
10 1 6 2 9 fvmptd3
` |-  ( ph -> ( C ` N ) = { c e. ( ( 0 ... N ) ^m ( 0 ... M ) ) | sum_ j e. ( 0 ... M ) ( c ` j ) = N } )`