Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1124.4 |
⊢ ( 𝜃 ↔ ( 𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ) ) |
2 |
|
bnj1124.5 |
⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) ) |
3 |
|
biid |
⊢ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
4 |
|
biid |
⊢ ( ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
5 |
|
biid |
⊢ ( ( 𝑛 ∈ ( ω ∖ { ∅ } ) ∧ 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ↔ ( 𝑛 ∈ ( ω ∖ { ∅ } ) ∧ 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
6 |
|
biid |
⊢ ( ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) |
7 |
|
eqid |
⊢ ( ω ∖ { ∅ } ) = ( ω ∖ { ∅ } ) |
8 |
|
eqid |
⊢ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } = { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } |
9 |
|
biid |
⊢ ( ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ↔ ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) |
10 |
|
biid |
⊢ ( ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) ↔ ∀ 𝑗 ∈ 𝑛 ( 𝑗 E 𝑖 → [ 𝑗 / 𝑖 ] ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) ) |
11 |
|
biid |
⊢ ( [ 𝑗 / 𝑖 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ [ 𝑗 / 𝑖 ] ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
12 |
|
biid |
⊢ ( [ 𝑗 / 𝑖 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ↔ [ 𝑗 / 𝑖 ] ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) |
13 |
|
biid |
⊢ ( [ 𝑗 / 𝑖 ] ( 𝑛 ∈ ( ω ∖ { ∅ } ) ∧ 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ↔ [ 𝑗 / 𝑖 ] ( 𝑛 ∈ ( ω ∖ { ∅ } ) ∧ 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) ) |
14 |
|
biid |
⊢ ( [ 𝑗 / 𝑖 ] 𝜃 ↔ [ 𝑗 / 𝑖 ] 𝜃 ) |
15 |
|
biid |
⊢ ( [ 𝑗 / 𝑖 ] 𝜏 ↔ [ 𝑗 / 𝑖 ] 𝜏 ) |
16 |
|
biid |
⊢ ( [ 𝑗 / 𝑖 ] ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ↔ [ 𝑗 / 𝑖 ] ( 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ ( 𝑓 ‘ 𝑖 ) ) ) |
17 |
|
biid |
⊢ ( [ 𝑗 / 𝑖 ] ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ↔ [ 𝑗 / 𝑖 ] ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) |
18 |
|
biid |
⊢ ( ( ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) → [ 𝑗 / 𝑖 ] ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) ↔ ( ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) → [ 𝑗 / 𝑖 ] ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) ) |
19 |
|
biid |
⊢ ( ( 𝑖 ∈ 𝑛 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) → [ 𝑗 / 𝑖 ] ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) ∧ 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) ↔ ( 𝑖 ∈ 𝑛 ∧ ( ( 𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖 ) → [ 𝑗 / 𝑖 ] ( ( 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) ) ∧ 𝑓 ∈ { 𝑓 ∣ ∃ 𝑛 ∈ ( ω ∖ { ∅ } ) ( 𝑓 Fn 𝑛 ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ ∀ 𝑖 ∈ ω ( suc 𝑖 ∈ 𝑛 → ( 𝑓 ‘ suc 𝑖 ) = ∪ 𝑦 ∈ ( 𝑓 ‘ 𝑖 ) pred ( 𝑦 , 𝐴 , 𝑅 ) ) ) } ∧ 𝑖 ∈ dom 𝑓 ) ) |
20 |
3 4 5 1 2 6 7 8 9 10 11 12 13 14 15 16 17 18 19
|
bnj1030 |
⊢ ( ( 𝜃 ∧ 𝜏 ) → trCl ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) |