Step |
Hyp |
Ref |
Expression |
1 |
|
pmtrrn.t |
⊢ 𝑇 = ( pmTrsp ‘ 𝐷 ) |
2 |
|
pmtrrn.r |
⊢ 𝑅 = ran 𝑇 |
3 |
|
eqid |
⊢ dom ( 𝐹 ∖ I ) = dom ( 𝐹 ∖ I ) |
4 |
1 2 3
|
pmtrfrn |
⊢ ( 𝐹 ∈ 𝑅 → ( ( 𝐷 ∈ V ∧ dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) ∧ 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) ) |
5 |
4
|
simpld |
⊢ ( 𝐹 ∈ 𝑅 → ( 𝐷 ∈ V ∧ dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ dom ( 𝐹 ∖ I ) ≈ 2o ) ) |
6 |
5
|
simp3d |
⊢ ( 𝐹 ∈ 𝑅 → dom ( 𝐹 ∖ I ) ≈ 2o ) |
7 |
|
en2 |
⊢ ( dom ( 𝐹 ∖ I ) ≈ 2o → ∃ 𝑥 ∃ 𝑦 dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } ) |
8 |
6 7
|
syl |
⊢ ( 𝐹 ∈ 𝑅 → ∃ 𝑥 ∃ 𝑦 dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } ) |
9 |
5
|
simp2d |
⊢ ( 𝐹 ∈ 𝑅 → dom ( 𝐹 ∖ I ) ⊆ 𝐷 ) |
10 |
4
|
simprd |
⊢ ( 𝐹 ∈ 𝑅 → 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) |
11 |
9 6 10
|
jca32 |
⊢ ( 𝐹 ∈ 𝑅 → ( dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ ( dom ( 𝐹 ∖ I ) ≈ 2o ∧ 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) ) ) |
12 |
|
sseq1 |
⊢ ( dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ( dom ( 𝐹 ∖ I ) ⊆ 𝐷 ↔ { 𝑥 , 𝑦 } ⊆ 𝐷 ) ) |
13 |
|
breq1 |
⊢ ( dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ( dom ( 𝐹 ∖ I ) ≈ 2o ↔ { 𝑥 , 𝑦 } ≈ 2o ) ) |
14 |
|
fveq2 |
⊢ ( dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) |
15 |
14
|
eqeq2d |
⊢ ( dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ( 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ↔ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) |
16 |
13 15
|
anbi12d |
⊢ ( dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ( ( dom ( 𝐹 ∖ I ) ≈ 2o ∧ 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) ↔ ( { 𝑥 , 𝑦 } ≈ 2o ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ) |
17 |
12 16
|
anbi12d |
⊢ ( dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ( ( dom ( 𝐹 ∖ I ) ⊆ 𝐷 ∧ ( dom ( 𝐹 ∖ I ) ≈ 2o ∧ 𝐹 = ( 𝑇 ‘ dom ( 𝐹 ∖ I ) ) ) ) ↔ ( { 𝑥 , 𝑦 } ⊆ 𝐷 ∧ ( { 𝑥 , 𝑦 } ≈ 2o ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ) ) |
18 |
11 17
|
syl5ibcom |
⊢ ( 𝐹 ∈ 𝑅 → ( dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ( { 𝑥 , 𝑦 } ⊆ 𝐷 ∧ ( { 𝑥 , 𝑦 } ≈ 2o ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ) ) |
19 |
|
vex |
⊢ 𝑥 ∈ V |
20 |
|
vex |
⊢ 𝑦 ∈ V |
21 |
19 20
|
prss |
⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ↔ { 𝑥 , 𝑦 } ⊆ 𝐷 ) |
22 |
21
|
bicomi |
⊢ ( { 𝑥 , 𝑦 } ⊆ 𝐷 ↔ ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ) |
23 |
|
pr2ne |
⊢ ( ( 𝑥 ∈ V ∧ 𝑦 ∈ V ) → ( { 𝑥 , 𝑦 } ≈ 2o ↔ 𝑥 ≠ 𝑦 ) ) |
24 |
23
|
el2v |
⊢ ( { 𝑥 , 𝑦 } ≈ 2o ↔ 𝑥 ≠ 𝑦 ) |
25 |
24
|
anbi1i |
⊢ ( ( { 𝑥 , 𝑦 } ≈ 2o ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ↔ ( 𝑥 ≠ 𝑦 ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) |
26 |
22 25
|
anbi12i |
⊢ ( ( { 𝑥 , 𝑦 } ⊆ 𝐷 ∧ ( { 𝑥 , 𝑦 } ≈ 2o ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ↔ ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ≠ 𝑦 ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ) |
27 |
18 26
|
syl6ib |
⊢ ( 𝐹 ∈ 𝑅 → ( dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ≠ 𝑦 ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ) ) |
28 |
27
|
2eximdv |
⊢ ( 𝐹 ∈ 𝑅 → ( ∃ 𝑥 ∃ 𝑦 dom ( 𝐹 ∖ I ) = { 𝑥 , 𝑦 } → ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ≠ 𝑦 ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ) ) |
29 |
8 28
|
mpd |
⊢ ( 𝐹 ∈ 𝑅 → ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ≠ 𝑦 ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ) |
30 |
|
r2ex |
⊢ ( ∃ 𝑥 ∈ 𝐷 ∃ 𝑦 ∈ 𝐷 ( 𝑥 ≠ 𝑦 ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ↔ ∃ 𝑥 ∃ 𝑦 ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ≠ 𝑦 ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) ) |
31 |
29 30
|
sylibr |
⊢ ( 𝐹 ∈ 𝑅 → ∃ 𝑥 ∈ 𝐷 ∃ 𝑦 ∈ 𝐷 ( 𝑥 ≠ 𝑦 ∧ 𝐹 = ( 𝑇 ‘ { 𝑥 , 𝑦 } ) ) ) |