Step |
Hyp |
Ref |
Expression |
1 |
|
ffn |
|- ( F : ( 0 ... M ) --> X -> F Fn ( 0 ... M ) ) |
2 |
|
ffn |
|- ( P : ( 0 ... N ) --> Y -> P Fn ( 0 ... N ) ) |
3 |
1 2
|
anim12i |
|- ( ( F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) ) |
4 |
3
|
3adant1 |
|- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) ) |
5 |
|
eqfnfv2 |
|- ( ( F Fn ( 0 ... M ) /\ P Fn ( 0 ... N ) ) -> ( F = P <-> ( ( 0 ... M ) = ( 0 ... N ) /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) ) |
6 |
4 5
|
syl |
|- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F = P <-> ( ( 0 ... M ) = ( 0 ... N ) /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) ) |
7 |
|
elnn0uz |
|- ( M e. NN0 <-> M e. ( ZZ>= ` 0 ) ) |
8 |
|
fzopth |
|- ( M e. ( ZZ>= ` 0 ) -> ( ( 0 ... M ) = ( 0 ... N ) <-> ( 0 = 0 /\ M = N ) ) ) |
9 |
7 8
|
sylbi |
|- ( M e. NN0 -> ( ( 0 ... M ) = ( 0 ... N ) <-> ( 0 = 0 /\ M = N ) ) ) |
10 |
|
simpr |
|- ( ( 0 = 0 /\ M = N ) -> M = N ) |
11 |
9 10
|
syl6bi |
|- ( M e. NN0 -> ( ( 0 ... M ) = ( 0 ... N ) -> M = N ) ) |
12 |
11
|
anim1d |
|- ( M e. NN0 -> ( ( ( 0 ... M ) = ( 0 ... N ) /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) -> ( M = N /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) ) |
13 |
|
oveq2 |
|- ( M = N -> ( 0 ... M ) = ( 0 ... N ) ) |
14 |
13
|
anim1i |
|- ( ( M = N /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) -> ( ( 0 ... M ) = ( 0 ... N ) /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) |
15 |
12 14
|
impbid1 |
|- ( M e. NN0 -> ( ( ( 0 ... M ) = ( 0 ... N ) /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) ) |
16 |
15
|
3ad2ant1 |
|- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( ( ( 0 ... M ) = ( 0 ... N ) /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) <-> ( M = N /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) ) |
17 |
6 16
|
bitrd |
|- ( ( M e. NN0 /\ F : ( 0 ... M ) --> X /\ P : ( 0 ... N ) --> Y ) -> ( F = P <-> ( M = N /\ A. i e. ( 0 ... M ) ( F ` i ) = ( P ` i ) ) ) ) |