Proof of Theorem bnj1097
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | bnj1097.3 | . . . . . . . 8
⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) | 
| 2 |  | bnj1097.1 | . . . . . . . . 9
⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| 3 | 2 | biimpi 216 | . . . . . . . 8
⊢ (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| 4 | 1, 3 | bnj771 34778 | . . . . . . 7
⊢ (𝜒 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| 5 | 4 | 3ad2ant3 1136 | . . . . . 6
⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| 6 | 5 | adantr 480 | . . . . 5
⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) | 
| 7 |  | bnj1097.5 | . . . . . . . 8
⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) | 
| 8 | 7 | simp3bi 1148 | . . . . . . 7
⊢ (𝜏 → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) | 
| 9 | 8 | 3ad2ant2 1135 | . . . . . 6
⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) | 
| 10 | 9 | adantr 480 | . . . . 5
⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) | 
| 11 | 6, 10 | jca 511 | . . . 4
⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) | 
| 12 | 11 | anim2i 617 | . . 3
⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))) | 
| 13 |  | 3anass 1095 | . . 3
⊢ ((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 234 | . 2
⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) | 
| 15 |  | fveqeq2 6915 | . . . . . 6
⊢ (𝑖 = ∅ → ((𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))) | 
| 16 | 15 | biimpar 477 | . . . . 5
⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅)) | 
| 17 | 16 | adantr 480 | . . . 4
⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅)) | 
| 18 |  | simpr 484 | . . . 4
⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) | 
| 19 | 17, 18 | eqsstrd 4018 | . . 3
⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) ⊆ 𝐵) | 
| 20 | 19 | 3impa 1110 | . 2
⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) ⊆ 𝐵) | 
| 21 | 14, 20 | syl 17 | 1
⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵) |