Step |
Hyp |
Ref |
Expression |
1 |
|
pmtrdifel.t |
⊢ 𝑇 = ran ( pmTrsp ‘ ( 𝑁 ∖ { 𝐾 } ) ) |
2 |
|
pmtrdifel.r |
⊢ 𝑅 = ran ( pmTrsp ‘ 𝑁 ) |
3 |
|
pmtrdifel.0 |
⊢ 𝑆 = ( ( pmTrsp ‘ 𝑁 ) ‘ dom ( 𝑄 ∖ I ) ) |
4 |
|
eqid |
⊢ ( pmTrsp ‘ ( 𝑁 ∖ { 𝐾 } ) ) = ( pmTrsp ‘ ( 𝑁 ∖ { 𝐾 } ) ) |
5 |
4 1
|
pmtrfb |
⊢ ( 𝑄 ∈ 𝑇 ↔ ( ( 𝑁 ∖ { 𝐾 } ) ∈ V ∧ 𝑄 : ( 𝑁 ∖ { 𝐾 } ) –1-1-onto→ ( 𝑁 ∖ { 𝐾 } ) ∧ dom ( 𝑄 ∖ I ) ≈ 2o ) ) |
6 |
|
difsnexi |
⊢ ( ( 𝑁 ∖ { 𝐾 } ) ∈ V → 𝑁 ∈ V ) |
7 |
|
f1of |
⊢ ( 𝑄 : ( 𝑁 ∖ { 𝐾 } ) –1-1-onto→ ( 𝑁 ∖ { 𝐾 } ) → 𝑄 : ( 𝑁 ∖ { 𝐾 } ) ⟶ ( 𝑁 ∖ { 𝐾 } ) ) |
8 |
|
fdm |
⊢ ( 𝑄 : ( 𝑁 ∖ { 𝐾 } ) ⟶ ( 𝑁 ∖ { 𝐾 } ) → dom 𝑄 = ( 𝑁 ∖ { 𝐾 } ) ) |
9 |
|
difssd |
⊢ ( dom 𝑄 = ( 𝑁 ∖ { 𝐾 } ) → ( 𝑄 ∖ I ) ⊆ 𝑄 ) |
10 |
|
dmss |
⊢ ( ( 𝑄 ∖ I ) ⊆ 𝑄 → dom ( 𝑄 ∖ I ) ⊆ dom 𝑄 ) |
11 |
9 10
|
syl |
⊢ ( dom 𝑄 = ( 𝑁 ∖ { 𝐾 } ) → dom ( 𝑄 ∖ I ) ⊆ dom 𝑄 ) |
12 |
|
difssd |
⊢ ( dom 𝑄 = ( 𝑁 ∖ { 𝐾 } ) → ( 𝑁 ∖ { 𝐾 } ) ⊆ 𝑁 ) |
13 |
|
sseq1 |
⊢ ( dom 𝑄 = ( 𝑁 ∖ { 𝐾 } ) → ( dom 𝑄 ⊆ 𝑁 ↔ ( 𝑁 ∖ { 𝐾 } ) ⊆ 𝑁 ) ) |
14 |
12 13
|
mpbird |
⊢ ( dom 𝑄 = ( 𝑁 ∖ { 𝐾 } ) → dom 𝑄 ⊆ 𝑁 ) |
15 |
11 14
|
sstrd |
⊢ ( dom 𝑄 = ( 𝑁 ∖ { 𝐾 } ) → dom ( 𝑄 ∖ I ) ⊆ 𝑁 ) |
16 |
7 8 15
|
3syl |
⊢ ( 𝑄 : ( 𝑁 ∖ { 𝐾 } ) –1-1-onto→ ( 𝑁 ∖ { 𝐾 } ) → dom ( 𝑄 ∖ I ) ⊆ 𝑁 ) |
17 |
|
id |
⊢ ( dom ( 𝑄 ∖ I ) ≈ 2o → dom ( 𝑄 ∖ I ) ≈ 2o ) |
18 |
|
eqid |
⊢ ( pmTrsp ‘ 𝑁 ) = ( pmTrsp ‘ 𝑁 ) |
19 |
18 2
|
pmtrrn |
⊢ ( ( 𝑁 ∈ V ∧ dom ( 𝑄 ∖ I ) ⊆ 𝑁 ∧ dom ( 𝑄 ∖ I ) ≈ 2o ) → ( ( pmTrsp ‘ 𝑁 ) ‘ dom ( 𝑄 ∖ I ) ) ∈ 𝑅 ) |
20 |
3 19
|
eqeltrid |
⊢ ( ( 𝑁 ∈ V ∧ dom ( 𝑄 ∖ I ) ⊆ 𝑁 ∧ dom ( 𝑄 ∖ I ) ≈ 2o ) → 𝑆 ∈ 𝑅 ) |
21 |
6 16 17 20
|
syl3an |
⊢ ( ( ( 𝑁 ∖ { 𝐾 } ) ∈ V ∧ 𝑄 : ( 𝑁 ∖ { 𝐾 } ) –1-1-onto→ ( 𝑁 ∖ { 𝐾 } ) ∧ dom ( 𝑄 ∖ I ) ≈ 2o ) → 𝑆 ∈ 𝑅 ) |
22 |
5 21
|
sylbi |
⊢ ( 𝑄 ∈ 𝑇 → 𝑆 ∈ 𝑅 ) |