Step |
Hyp |
Ref |
Expression |
1 |
|
cnvco |
⊢ ◡ ( 𝐹 ∘ 𝐺 ) = ( ◡ 𝐺 ∘ ◡ 𝐹 ) |
2 |
1
|
imaeq1i |
⊢ ( ◡ ( 𝐹 ∘ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) = ( ( ◡ 𝐺 ∘ ◡ 𝐹 ) “ ( V ∖ { 𝑍 } ) ) |
3 |
|
imaco |
⊢ ( ( ◡ 𝐺 ∘ ◡ 𝐹 ) “ ( V ∖ { 𝑍 } ) ) = ( ◡ 𝐺 “ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
4 |
2 3
|
eqtri |
⊢ ( ◡ ( 𝐹 ∘ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) = ( ◡ 𝐺 “ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
5 |
4
|
imaeq2i |
⊢ ( 𝐺 “ ( ◡ ( 𝐹 ∘ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) ) = ( 𝐺 “ ( ◡ 𝐺 “ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ) |
6 |
|
funforn |
⊢ ( Fun 𝐺 ↔ 𝐺 : dom 𝐺 –onto→ ran 𝐺 ) |
7 |
6
|
biimpi |
⊢ ( Fun 𝐺 → 𝐺 : dom 𝐺 –onto→ ran 𝐺 ) |
8 |
7
|
ad2antrl |
⊢ ( ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) ∧ ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) ) → 𝐺 : dom 𝐺 –onto→ ran 𝐺 ) |
9 |
|
simpl |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) → 𝐹 ∈ 𝑉 ) |
10 |
9
|
anim2i |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( 𝑍 ∈ V ∧ 𝐹 ∈ 𝑉 ) ) |
11 |
10
|
ancomd |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V ) ) |
12 |
|
suppimacnv |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝑍 ∈ V ) → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
13 |
11 12
|
syl |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
14 |
13
|
sseq1d |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ↔ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ran 𝐺 ) ) |
15 |
14
|
biimpd |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 → ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ran 𝐺 ) ) |
16 |
15
|
adantld |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) → ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ran 𝐺 ) ) |
17 |
16
|
imp |
⊢ ( ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) ∧ ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) ) → ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ran 𝐺 ) |
18 |
|
foimacnv |
⊢ ( ( 𝐺 : dom 𝐺 –onto→ ran 𝐺 ∧ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ⊆ ran 𝐺 ) → ( 𝐺 “ ( ◡ 𝐺 “ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
19 |
8 17 18
|
syl2anc |
⊢ ( ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) ∧ ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) ) → ( 𝐺 “ ( ◡ 𝐺 “ ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
20 |
5 19
|
syl5eq |
⊢ ( ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) ∧ ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) ) → ( 𝐺 “ ( ◡ ( 𝐹 ∘ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
21 |
|
coexg |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) → ( 𝐹 ∘ 𝐺 ) ∈ V ) |
22 |
21
|
anim2i |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( 𝑍 ∈ V ∧ ( 𝐹 ∘ 𝐺 ) ∈ V ) ) |
23 |
22
|
ancomd |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( ( 𝐹 ∘ 𝐺 ) ∈ V ∧ 𝑍 ∈ V ) ) |
24 |
|
suppimacnv |
⊢ ( ( ( 𝐹 ∘ 𝐺 ) ∈ V ∧ 𝑍 ∈ V ) → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) = ( ◡ ( 𝐹 ∘ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) ) |
25 |
23 24
|
syl |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) = ( ◡ ( 𝐹 ∘ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) ) |
26 |
25
|
imaeq2d |
⊢ ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) → ( 𝐺 “ ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ) = ( 𝐺 “ ( ◡ ( 𝐹 ∘ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) ) ) |
27 |
26
|
adantr |
⊢ ( ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) ∧ ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) ) → ( 𝐺 “ ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ) = ( 𝐺 “ ( ◡ ( 𝐹 ∘ 𝐺 ) “ ( V ∖ { 𝑍 } ) ) ) ) |
28 |
13
|
adantr |
⊢ ( ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) ∧ ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) ) → ( 𝐹 supp 𝑍 ) = ( ◡ 𝐹 “ ( V ∖ { 𝑍 } ) ) ) |
29 |
20 27 28
|
3eqtr4d |
⊢ ( ( ( 𝑍 ∈ V ∧ ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) ) ∧ ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) ) → ( 𝐺 “ ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ) = ( 𝐹 supp 𝑍 ) ) |
30 |
29
|
exp31 |
⊢ ( 𝑍 ∈ V → ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) → ( ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) → ( 𝐺 “ ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ) = ( 𝐹 supp 𝑍 ) ) ) ) |
31 |
|
ima0 |
⊢ ( 𝐺 “ ∅ ) = ∅ |
32 |
|
id |
⊢ ( ¬ 𝑍 ∈ V → ¬ 𝑍 ∈ V ) |
33 |
32
|
intnand |
⊢ ( ¬ 𝑍 ∈ V → ¬ ( ( 𝐹 ∘ 𝐺 ) ∈ V ∧ 𝑍 ∈ V ) ) |
34 |
|
supp0prc |
⊢ ( ¬ ( ( 𝐹 ∘ 𝐺 ) ∈ V ∧ 𝑍 ∈ V ) → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) = ∅ ) |
35 |
33 34
|
syl |
⊢ ( ¬ 𝑍 ∈ V → ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) = ∅ ) |
36 |
35
|
imaeq2d |
⊢ ( ¬ 𝑍 ∈ V → ( 𝐺 “ ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ) = ( 𝐺 “ ∅ ) ) |
37 |
32
|
intnand |
⊢ ( ¬ 𝑍 ∈ V → ¬ ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) ) |
38 |
|
supp0prc |
⊢ ( ¬ ( 𝐹 ∈ V ∧ 𝑍 ∈ V ) → ( 𝐹 supp 𝑍 ) = ∅ ) |
39 |
37 38
|
syl |
⊢ ( ¬ 𝑍 ∈ V → ( 𝐹 supp 𝑍 ) = ∅ ) |
40 |
31 36 39
|
3eqtr4a |
⊢ ( ¬ 𝑍 ∈ V → ( 𝐺 “ ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ) = ( 𝐹 supp 𝑍 ) ) |
41 |
40
|
2a1d |
⊢ ( ¬ 𝑍 ∈ V → ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) → ( ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) → ( 𝐺 “ ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ) = ( 𝐹 supp 𝑍 ) ) ) ) |
42 |
30 41
|
pm2.61i |
⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑊 ) → ( ( Fun 𝐺 ∧ ( 𝐹 supp 𝑍 ) ⊆ ran 𝐺 ) → ( 𝐺 “ ( ( 𝐹 ∘ 𝐺 ) supp 𝑍 ) ) = ( 𝐹 supp 𝑍 ) ) ) |