| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | uspgrsprf.p | . . . 4
⊢ 𝑃 = 𝒫 (Pairs‘𝑉) | 
| 2 |  | uspgrsprf.g | . . . 4
⊢ 𝐺 = {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} | 
| 3 |  | uspgrsprf.f | . . . 4
⊢ 𝐹 = (𝑔 ∈ 𝐺 ↦ (2nd ‘𝑔)) | 
| 4 | 1, 2, 3 | uspgrsprf 48067 | . . 3
⊢ 𝐹:𝐺⟶𝑃 | 
| 5 | 4 | a1i 11 | . 2
⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺⟶𝑃) | 
| 6 | 1 | eleq2i 2832 | . . . . . . 7
⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ∈ 𝒫 (Pairs‘𝑉)) | 
| 7 |  | velpw 4604 | . . . . . . 7
⊢ (𝑎 ∈ 𝒫
(Pairs‘𝑉) ↔
𝑎 ⊆
(Pairs‘𝑉)) | 
| 8 | 6, 7 | bitri 275 | . . . . . 6
⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ⊆ (Pairs‘𝑉)) | 
| 9 |  | eqidd 2737 | . . . . . . . . . 10
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑉 = 𝑉) | 
| 10 |  | vex 3483 | . . . . . . . . . . . . . . 15
⊢ 𝑎 ∈ V | 
| 11 | 10 | a1i 11 | . . . . . . . . . . . . . 14
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ∈ V) | 
| 12 |  | f1oi 6885 | . . . . . . . . . . . . . . . . 17
⊢ ( I
↾ 𝑎):𝑎–1-1-onto→𝑎 | 
| 13 | 12 | a1i 11 | . . . . . . . . . . . . . . . 16
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):𝑎–1-1-onto→𝑎) | 
| 14 |  | dmresi 6069 | . . . . . . . . . . . . . . . . 17
⊢ dom ( I
↾ 𝑎) = 𝑎 | 
| 15 |  | f1oeq2 6836 | . . . . . . . . . . . . . . . . 17
⊢ (dom ( I
↾ 𝑎) = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ↔ ( I ↾ 𝑎):𝑎–1-1-onto→𝑎)) | 
| 16 | 14, 15 | ax-mp 5 | . . . . . . . . . . . . . . . 16
⊢ (( I
↾ 𝑎):dom ( I ↾
𝑎)–1-1-onto→𝑎 ↔ ( I ↾ 𝑎):𝑎–1-1-onto→𝑎) | 
| 17 | 13, 16 | sylibr 234 | . . . . . . . . . . . . . . 15
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎) | 
| 18 |  | sprvalpwle2 47481 | . . . . . . . . . . . . . . . . 17
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2}) | 
| 19 | 18 | sseq2d 4015 | . . . . . . . . . . . . . . . 16
⊢ (𝑉 ∈ 𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) | 
| 20 | 19 | biimpac 478 | . . . . . . . . . . . . . . 15
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2}) | 
| 21 | 17, 20 | jca 511 | . . . . . . . . . . . . . 14
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ∧ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) | 
| 22 |  | f1oeq3 6837 | . . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎)) | 
| 23 |  | sseq1 4008 | . . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝑎 → (𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤ 2}
↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) | 
| 24 | 22, 23 | anbi12d 632 | . . . . . . . . . . . . . 14
⊢ (𝑓 = 𝑎 → ((( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤ 2})
↔ (( I ↾ 𝑎):dom
( I ↾ 𝑎)–1-1-onto→𝑎 ∧ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2}))) | 
| 25 | 11, 21, 24 | spcedv 3597 | . . . . . . . . . . . . 13
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) | 
| 26 |  | resiexg 7935 | . . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ V → ( I ↾
𝑎) ∈
V) | 
| 27 | 10, 26 | ax-mp 5 | . . . . . . . . . . . . . 14
⊢ ( I
↾ 𝑎) ∈
V | 
| 28 | 27 | f11o 7972 | . . . . . . . . . . . . 13
⊢ (( I
↾ 𝑎):dom ( I ↾
𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤ 2}
↔ ∃𝑓(( I ↾
𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) | 
| 29 | 25, 28 | sylibr 234 | . . . . . . . . . . . 12
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2}) | 
| 30 | 10 | a1i 11 | . . . . . . . . . . . . . . 15
⊢ (𝑎 ⊆ (Pairs‘𝑉) → 𝑎 ∈ V) | 
| 31 | 30 | resiexd 7237 | . . . . . . . . . . . . . 14
⊢ (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V) | 
| 32 | 31 | anim1ci 616 | . . . . . . . . . . . . 13
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V)) | 
| 33 |  | isuspgrop 29179 | . . . . . . . . . . . . 13
⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (〈𝑉, ( I ↾ 𝑎)〉 ∈ USPGraph ↔ ( I ↾
𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) | 
| 34 | 32, 33 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (〈𝑉, ( I ↾ 𝑎)〉 ∈ USPGraph ↔ ( I ↾
𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) | 
| 35 | 29, 34 | mpbird 257 | . . . . . . . . . . 11
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 〈𝑉, ( I ↾ 𝑎)〉 ∈ USPGraph) | 
| 36 |  | fveqeq2 6914 | . . . . . . . . . . . . 13
⊢ (𝑞 = 〈𝑉, ( I ↾ 𝑎)〉 → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉)) | 
| 37 |  | fveqeq2 6914 | . . . . . . . . . . . . 13
⊢ (𝑞 = 〈𝑉, ( I ↾ 𝑎)〉 → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎)) | 
| 38 | 36, 37 | anbi12d 632 | . . . . . . . . . . . 12
⊢ (𝑞 = 〈𝑉, ( I ↾ 𝑎)〉 → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉 ∧ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎))) | 
| 39 | 38 | adantl 481 | . . . . . . . . . . 11
⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) ∧ 𝑞 = 〈𝑉, ( I ↾ 𝑎)〉) → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉 ∧ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎))) | 
| 40 |  | opvtxfv 29022 | . . . . . . . . . . . . . 14
⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉) | 
| 41 | 31, 40 | sylan2 593 | . . . . . . . . . . . . 13
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉) | 
| 42 |  | edgopval 29069 | . . . . . . . . . . . . . . 15
⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = ran ( I ↾ 𝑎)) | 
| 43 | 31, 42 | sylan2 593 | . . . . . . . . . . . . . 14
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = ran ( I ↾ 𝑎)) | 
| 44 |  | rnresi 6092 | . . . . . . . . . . . . . 14
⊢ ran ( I
↾ 𝑎) = 𝑎 | 
| 45 | 43, 44 | eqtrdi 2792 | . . . . . . . . . . . . 13
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎) | 
| 46 | 41, 45 | jca 511 | . . . . . . . . . . . 12
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉 ∧ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎)) | 
| 47 | 46 | ancoms 458 | . . . . . . . . . . 11
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ((Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉 ∧ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎)) | 
| 48 | 35, 39, 47 | rspcedvd 3623 | . . . . . . . . . 10
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)) | 
| 49 | 9, 48 | jca 511 | . . . . . . . . 9
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))) | 
| 50 | 2 | eleq2i 2832 | . . . . . . . . . 10
⊢
(〈𝑉, 𝑎〉 ∈ 𝐺 ↔ 〈𝑉, 𝑎〉 ∈ {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}) | 
| 51 | 30 | anim1ci 616 | . . . . . . . . . . 11
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V)) | 
| 52 |  | eqeq1 2740 | . . . . . . . . . . . . . 14
⊢ (𝑣 = 𝑉 → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉)) | 
| 53 | 52 | adantr 480 | . . . . . . . . . . . . 13
⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉)) | 
| 54 |  | eqeq2 2748 | . . . . . . . . . . . . . . 15
⊢ (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉)) | 
| 55 |  | eqeq2 2748 | . . . . . . . . . . . . . . 15
⊢ (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎)) | 
| 56 | 54, 55 | bi2anan9 638 | . . . . . . . . . . . . . 14
⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))) | 
| 57 | 56 | rexbidv 3178 | . . . . . . . . . . . . 13
⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))) | 
| 58 | 53, 57 | anbi12d 632 | . . . . . . . . . . . 12
⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → ((𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒)) ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))) | 
| 59 | 58 | opelopabga 5537 | . . . . . . . . . . 11
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V) → (〈𝑉, 𝑎〉 ∈ {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))) | 
| 60 | 51, 59 | syl 17 | . . . . . . . . . 10
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (〈𝑉, 𝑎〉 ∈ {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))} ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))) | 
| 61 | 50, 60 | bitrid 283 | . . . . . . . . 9
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (〈𝑉, 𝑎〉 ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))) | 
| 62 | 49, 61 | mpbird 257 | . . . . . . . 8
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 〈𝑉, 𝑎〉 ∈ 𝐺) | 
| 63 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑏 = 〈𝑉, 𝑎〉 → (2nd ‘𝑏) = (2nd
‘〈𝑉, 𝑎〉)) | 
| 64 | 63 | eqeq2d 2747 | . . . . . . . . 9
⊢ (𝑏 = 〈𝑉, 𝑎〉 → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘〈𝑉, 𝑎〉))) | 
| 65 | 64 | adantl 481 | . . . . . . . 8
⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) ∧ 𝑏 = 〈𝑉, 𝑎〉) → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘〈𝑉, 𝑎〉))) | 
| 66 |  | op2ndg 8028 | . . . . . . . . . . 11
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V) → (2nd
‘〈𝑉, 𝑎〉) = 𝑎) | 
| 67 | 66 | elvd 3485 | . . . . . . . . . 10
⊢ (𝑉 ∈ 𝑊 → (2nd ‘〈𝑉, 𝑎〉) = 𝑎) | 
| 68 | 67 | adantl 481 | . . . . . . . . 9
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (2nd ‘〈𝑉, 𝑎〉) = 𝑎) | 
| 69 | 68 | eqcomd 2742 | . . . . . . . 8
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 = (2nd ‘〈𝑉, 𝑎〉)) | 
| 70 | 62, 65, 69 | rspcedvd 3623 | . . . . . . 7
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)) | 
| 71 | 70 | ex 412 | . . . . . 6
⊢ (𝑎 ⊆ (Pairs‘𝑉) → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))) | 
| 72 | 8, 71 | sylbi 217 | . . . . 5
⊢ (𝑎 ∈ 𝑃 → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))) | 
| 73 | 72 | impcom 407 | . . . 4
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)) | 
| 74 | 1, 2, 3 | uspgrsprfv 48066 | . . . . . . 7
⊢ (𝑏 ∈ 𝐺 → (𝐹‘𝑏) = (2nd ‘𝑏)) | 
| 75 | 74 | adantl 481 | . . . . . 6
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝐹‘𝑏) = (2nd ‘𝑏)) | 
| 76 | 75 | eqeq2d 2747 | . . . . 5
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝑎 = (𝐹‘𝑏) ↔ 𝑎 = (2nd ‘𝑏))) | 
| 77 | 76 | rexbidva 3176 | . . . 4
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → (∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏) ↔ ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))) | 
| 78 | 73, 77 | mpbird 257 | . . 3
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏)) | 
| 79 | 78 | ralrimiva 3145 | . 2
⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏)) | 
| 80 |  | dffo3 7121 | . 2
⊢ (𝐹:𝐺–onto→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏))) | 
| 81 | 5, 79, 80 | sylanbrc 583 | 1
⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃) |