Step |
Hyp |
Ref |
Expression |
1 |
|
txcmpb.1 |
|- X = U. R |
2 |
|
txcmpb.2 |
|- Y = U. S |
3 |
|
simpr |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( R tX S ) e. Comp ) |
4 |
|
simplrr |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> Y =/= (/) ) |
5 |
|
fo1stres |
|- ( Y =/= (/) -> ( 1st |` ( X X. Y ) ) : ( X X. Y ) -onto-> X ) |
6 |
4 5
|
syl |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 1st |` ( X X. Y ) ) : ( X X. Y ) -onto-> X ) |
7 |
1 2
|
txuni |
|- ( ( R e. Top /\ S e. Top ) -> ( X X. Y ) = U. ( R tX S ) ) |
8 |
7
|
ad2antrr |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( X X. Y ) = U. ( R tX S ) ) |
9 |
|
foeq2 |
|- ( ( X X. Y ) = U. ( R tX S ) -> ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) -onto-> X <-> ( 1st |` ( X X. Y ) ) : U. ( R tX S ) -onto-> X ) ) |
10 |
8 9
|
syl |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( ( 1st |` ( X X. Y ) ) : ( X X. Y ) -onto-> X <-> ( 1st |` ( X X. Y ) ) : U. ( R tX S ) -onto-> X ) ) |
11 |
6 10
|
mpbid |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 1st |` ( X X. Y ) ) : U. ( R tX S ) -onto-> X ) |
12 |
1
|
toptopon |
|- ( R e. Top <-> R e. ( TopOn ` X ) ) |
13 |
2
|
toptopon |
|- ( S e. Top <-> S e. ( TopOn ` Y ) ) |
14 |
|
tx1cn |
|- ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) ) |
15 |
12 13 14
|
syl2anb |
|- ( ( R e. Top /\ S e. Top ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) ) |
16 |
15
|
ad2antrr |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) ) |
17 |
1
|
cncmp |
|- ( ( ( R tX S ) e. Comp /\ ( 1st |` ( X X. Y ) ) : U. ( R tX S ) -onto-> X /\ ( 1st |` ( X X. Y ) ) e. ( ( R tX S ) Cn R ) ) -> R e. Comp ) |
18 |
3 11 16 17
|
syl3anc |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> R e. Comp ) |
19 |
|
simplrl |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> X =/= (/) ) |
20 |
|
fo2ndres |
|- ( X =/= (/) -> ( 2nd |` ( X X. Y ) ) : ( X X. Y ) -onto-> Y ) |
21 |
19 20
|
syl |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 2nd |` ( X X. Y ) ) : ( X X. Y ) -onto-> Y ) |
22 |
|
foeq2 |
|- ( ( X X. Y ) = U. ( R tX S ) -> ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) -onto-> Y <-> ( 2nd |` ( X X. Y ) ) : U. ( R tX S ) -onto-> Y ) ) |
23 |
8 22
|
syl |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( ( 2nd |` ( X X. Y ) ) : ( X X. Y ) -onto-> Y <-> ( 2nd |` ( X X. Y ) ) : U. ( R tX S ) -onto-> Y ) ) |
24 |
21 23
|
mpbid |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 2nd |` ( X X. Y ) ) : U. ( R tX S ) -onto-> Y ) |
25 |
|
tx2cn |
|- ( ( R e. ( TopOn ` X ) /\ S e. ( TopOn ` Y ) ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) ) |
26 |
12 13 25
|
syl2anb |
|- ( ( R e. Top /\ S e. Top ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) ) |
27 |
26
|
ad2antrr |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) ) |
28 |
2
|
cncmp |
|- ( ( ( R tX S ) e. Comp /\ ( 2nd |` ( X X. Y ) ) : U. ( R tX S ) -onto-> Y /\ ( 2nd |` ( X X. Y ) ) e. ( ( R tX S ) Cn S ) ) -> S e. Comp ) |
29 |
3 24 27 28
|
syl3anc |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> S e. Comp ) |
30 |
18 29
|
jca |
|- ( ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) /\ ( R tX S ) e. Comp ) -> ( R e. Comp /\ S e. Comp ) ) |
31 |
30
|
ex |
|- ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( ( R tX S ) e. Comp -> ( R e. Comp /\ S e. Comp ) ) ) |
32 |
|
txcmp |
|- ( ( R e. Comp /\ S e. Comp ) -> ( R tX S ) e. Comp ) |
33 |
31 32
|
impbid1 |
|- ( ( ( R e. Top /\ S e. Top ) /\ ( X =/= (/) /\ Y =/= (/) ) ) -> ( ( R tX S ) e. Comp <-> ( R e. Comp /\ S e. Comp ) ) ) |