| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elex 3500 | . 2
⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ V) | 
| 2 |  | relwdom 9607 | . . . . 5
⊢ Rel
≼* | 
| 3 | 2 | brrelex1i 5740 | . . . 4
⊢ (𝑋 ≼* 𝑌 → 𝑋 ∈ V) | 
| 4 | 3 | a1i 11 | . . 3
⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 → 𝑋 ∈ V)) | 
| 5 |  | 0ex 5306 | . . . . . 6
⊢ ∅
∈ V | 
| 6 |  | eleq1a 2835 | . . . . . 6
⊢ (∅
∈ V → (𝑋 =
∅ → 𝑋 ∈
V)) | 
| 7 | 5, 6 | ax-mp 5 | . . . . 5
⊢ (𝑋 = ∅ → 𝑋 ∈ V) | 
| 8 |  | forn 6822 | . . . . . . 7
⊢ (𝑧:𝑌–onto→𝑋 → ran 𝑧 = 𝑋) | 
| 9 |  | vex 3483 | . . . . . . . 8
⊢ 𝑧 ∈ V | 
| 10 | 9 | rnex 7933 | . . . . . . 7
⊢ ran 𝑧 ∈ V | 
| 11 | 8, 10 | eqeltrrdi 2849 | . . . . . 6
⊢ (𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V) | 
| 12 | 11 | exlimiv 1929 | . . . . 5
⊢
(∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V) | 
| 13 | 7, 12 | jaoi 857 | . . . 4
⊢ ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) → 𝑋 ∈ V) | 
| 14 | 13 | a1i 11 | . . 3
⊢ (𝑌 ∈ V → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) → 𝑋 ∈ V)) | 
| 15 |  | eqeq1 2740 | . . . . . 6
⊢ (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅)) | 
| 16 |  | foeq3 6817 | . . . . . . 7
⊢ (𝑥 = 𝑋 → (𝑧:𝑦–onto→𝑥 ↔ 𝑧:𝑦–onto→𝑋)) | 
| 17 | 16 | exbidv 1920 | . . . . . 6
⊢ (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦–onto→𝑥 ↔ ∃𝑧 𝑧:𝑦–onto→𝑋)) | 
| 18 | 15, 17 | orbi12d 918 | . . . . 5
⊢ (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋))) | 
| 19 |  | foeq2 6816 | . . . . . . 7
⊢ (𝑦 = 𝑌 → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋)) | 
| 20 | 19 | exbidv 1920 | . . . . . 6
⊢ (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋)) | 
| 21 | 20 | orbi2d 915 | . . . . 5
⊢ (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | 
| 22 |  | df-wdom 9606 | . . . . 5
⊢ 
≼* = {〈𝑥, 𝑦〉 ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)} | 
| 23 | 18, 21, 22 | brabg 5543 | . . . 4
⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | 
| 24 | 23 | expcom 413 | . . 3
⊢ (𝑌 ∈ V → (𝑋 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))) | 
| 25 | 4, 14, 24 | pm5.21ndd 379 | . 2
⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) | 
| 26 | 1, 25 | syl 17 | 1
⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))) |