Proof of Theorem bnj1366
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | bnj1366.1 | . . . 4
⊢ (𝜓 ↔ (𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 ∧ 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑})) | 
| 2 | 1 | simp3bi 1147 | . . 3
⊢ (𝜓 → 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑}) | 
| 3 | 1 | simp2bi 1146 | . . . . 5
⊢ (𝜓 → ∀𝑥 ∈ 𝐴 ∃!𝑦𝜑) | 
| 4 |  | nfcv 2904 | . . . . . . 7
⊢
Ⅎ𝑦𝐴 | 
| 5 |  | nfeu1 2587 | . . . . . . 7
⊢
Ⅎ𝑦∃!𝑦𝜑 | 
| 6 | 4, 5 | nfralw 3310 | . . . . . 6
⊢
Ⅎ𝑦∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 | 
| 7 |  | nfra1 3283 | . . . . . . . 8
⊢
Ⅎ𝑥∀𝑥 ∈ 𝐴 ∃!𝑦𝜑 | 
| 8 |  | rspa 3247 | . . . . . . . . 9
⊢
((∀𝑥 ∈
𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → ∃!𝑦𝜑) | 
| 9 |  | iota1 6537 | . . . . . . . . . 10
⊢
(∃!𝑦𝜑 → (𝜑 ↔ (℩𝑦𝜑) = 𝑦)) | 
| 10 |  | eqcom 2743 | . . . . . . . . . 10
⊢
((℩𝑦𝜑) = 𝑦 ↔ 𝑦 = (℩𝑦𝜑)) | 
| 11 | 9, 10 | bitrdi 287 | . . . . . . . . 9
⊢
(∃!𝑦𝜑 → (𝜑 ↔ 𝑦 = (℩𝑦𝜑))) | 
| 12 | 8, 11 | syl 17 | . . . . . . . 8
⊢
((∀𝑥 ∈
𝐴 ∃!𝑦𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜑 ↔ 𝑦 = (℩𝑦𝜑))) | 
| 13 | 7, 12 | rexbida 3271 | . . . . . . 7
⊢
(∀𝑥 ∈
𝐴 ∃!𝑦𝜑 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑))) | 
| 14 |  | abid 2717 | . . . . . . 7
⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ ∃𝑥 ∈ 𝐴 𝜑) | 
| 15 |  | eqid 2736 | . . . . . . . 8
⊢ (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) = (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) | 
| 16 |  | iotaex 6533 | . . . . . . . 8
⊢
(℩𝑦𝜑) ∈ V | 
| 17 | 15, 16 | elrnmpti 5972 | . . . . . . 7
⊢ (𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (℩𝑦𝜑)) | 
| 18 | 13, 14, 17 | 3bitr4g 314 | . . . . . 6
⊢
(∀𝑥 ∈
𝐴 ∃!𝑦𝜑 → (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))) | 
| 19 | 6, 18 | alrimi 2212 | . . . . 5
⊢
(∀𝑥 ∈
𝐴 ∃!𝑦𝜑 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))) | 
| 20 | 3, 19 | syl 17 | . . . 4
⊢ (𝜓 → ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))) | 
| 21 |  | nfab1 2906 | . . . . 5
⊢
Ⅎ𝑦{𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} | 
| 22 |  | nfiota1 6515 | . . . . . . 7
⊢
Ⅎ𝑦(℩𝑦𝜑) | 
| 23 | 4, 22 | nfmpt 5248 | . . . . . 6
⊢
Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) | 
| 24 | 23 | nfrn 5962 | . . . . 5
⊢
Ⅎ𝑦ran
(𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) | 
| 25 | 21, 24 | cleqf 2933 | . . . 4
⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ↔ ∀𝑦(𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ↔ 𝑦 ∈ ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)))) | 
| 26 | 20, 25 | sylibr 234 | . . 3
⊢ (𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))) | 
| 27 | 2, 26 | eqtrd 2776 | . 2
⊢ (𝜓 → 𝐵 = ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑))) | 
| 28 | 1 | simp1bi 1145 | . . 3
⊢ (𝜓 → 𝐴 ∈ V) | 
| 29 |  | mptexg 7242 | . . 3
⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V) | 
| 30 |  | rnexg 7925 | . . 3
⊢ ((𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V → ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V) | 
| 31 | 28, 29, 30 | 3syl 18 | . 2
⊢ (𝜓 → ran (𝑥 ∈ 𝐴 ↦ (℩𝑦𝜑)) ∈ V) | 
| 32 | 27, 31 | eqeltrd 2840 | 1
⊢ (𝜓 → 𝐵 ∈ V) |