Step |
Hyp |
Ref |
Expression |
1 |
|
bnj1097.1 |
⊢ ( 𝜑 ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
2 |
|
bnj1097.3 |
⊢ ( 𝜒 ↔ ( 𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓 ) ) |
3 |
|
bnj1097.5 |
⊢ ( 𝜏 ↔ ( 𝐵 ∈ V ∧ TrFo ( 𝐵 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) ) |
4 |
1
|
biimpi |
⊢ ( 𝜑 → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
5 |
2 4
|
bnj771 |
⊢ ( 𝜒 → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
6 |
5
|
3ad2ant3 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
7 |
6
|
adantr |
⊢ ( ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) → ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
8 |
3
|
simp3bi |
⊢ ( 𝜏 → pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) |
9 |
8
|
3ad2ant2 |
⊢ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) → pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) |
10 |
9
|
adantr |
⊢ ( ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) → pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) |
11 |
7 10
|
jca |
⊢ ( ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) → ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) ) |
12 |
11
|
anim2i |
⊢ ( ( 𝑖 = ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑖 = ∅ ∧ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) ) ) |
13 |
|
3anass |
⊢ ( ( 𝑖 = ∅ ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) ↔ ( 𝑖 = ∅ ∧ ( ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) ) ) |
14 |
12 13
|
sylibr |
⊢ ( ( 𝑖 = ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑖 = ∅ ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) ) |
15 |
|
fveqeq2 |
⊢ ( 𝑖 = ∅ → ( ( 𝑓 ‘ 𝑖 ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ↔ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) ) |
16 |
15
|
biimpar |
⊢ ( ( 𝑖 = ∅ ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) → ( 𝑓 ‘ 𝑖 ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
17 |
16
|
adantr |
⊢ ( ( ( 𝑖 = ∅ ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) → ( 𝑓 ‘ 𝑖 ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) |
18 |
|
simpr |
⊢ ( ( ( 𝑖 = ∅ ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) → pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) |
19 |
17 18
|
eqsstrd |
⊢ ( ( ( 𝑖 = ∅ ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) |
20 |
19
|
3impa |
⊢ ( ( 𝑖 = ∅ ∧ ( 𝑓 ‘ ∅ ) = pred ( 𝑋 , 𝐴 , 𝑅 ) ∧ pred ( 𝑋 , 𝐴 , 𝑅 ) ⊆ 𝐵 ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) |
21 |
14 20
|
syl |
⊢ ( ( 𝑖 = ∅ ∧ ( ( 𝜃 ∧ 𝜏 ∧ 𝜒 ) ∧ 𝜑0 ) ) → ( 𝑓 ‘ 𝑖 ) ⊆ 𝐵 ) |