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 44028 |
. . 3
⊢ 𝐹:𝐺⟶𝑃 |
5 | 4 | a1i 11 |
. 2
⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺⟶𝑃) |
6 | 1 | eleq2i 2907 |
. . . . . . 7
⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ∈ 𝒫 (Pairs‘𝑉)) |
7 | | velpw 4547 |
. . . . . . 7
⊢ (𝑎 ∈ 𝒫
(Pairs‘𝑉) ↔
𝑎 ⊆
(Pairs‘𝑉)) |
8 | 6, 7 | bitri 277 |
. . . . . 6
⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ⊆ (Pairs‘𝑉)) |
9 | | eqidd 2825 |
. . . . . . . . . 10
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑉 = 𝑉) |
10 | | vex 3500 |
. . . . . . . . . . . . . . 15
⊢ 𝑎 ∈ V |
11 | 10 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ∈ V) |
12 | | f1oi 6655 |
. . . . . . . . . . . . . . . . 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 5924 |
. . . . . . . . . . . . . . . . 17
⊢ dom ( I
↾ 𝑎) = 𝑎 |
15 | | f1oeq2 6608 |
. . . . . . . . . . . . . . . . 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 236 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎) |
18 | | sprvalpwle2 43658 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑉 ∈ 𝑊 → (Pairs‘𝑉) = {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2}) |
19 | 18 | sseq2d 4002 |
. . . . . . . . . . . . . . . 16
⊢ (𝑉 ∈ 𝑊 → (𝑎 ⊆ (Pairs‘𝑉) ↔ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) |
20 | 19 | biimpac 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2}) |
21 | 17, 20 | jca 514 |
. . . . . . . . . . . . . 14
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎 ∧ 𝑎 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) |
22 | | f1oeq3 6609 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = 𝑎 → (( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ↔ ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑎)) |
23 | | sseq1 3995 |
. . . . . . . . . . . . . . 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 3602 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑓(( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1-onto→𝑓 ∧ 𝑓 ⊆ {𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) |
26 | | resiexg 7622 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ∈ V → ( I ↾
𝑎) ∈
V) |
27 | 10, 26 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ( I
↾ 𝑎) ∈
V |
28 | 27 | f11o 7651 |
. . . . . . . . . . . . 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 236 |
. . . . . . . . . . . 12
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ( I ↾ 𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2}) |
30 | 10 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ⊆ (Pairs‘𝑉) → 𝑎 ∈ V) |
31 | 30 | resiexd 6982 |
. . . . . . . . . . . . . . 15
⊢ (𝑎 ⊆ (Pairs‘𝑉) → ( I ↾ 𝑎) ∈ V) |
32 | 31 | anim2i 618 |
. . . . . . . . . . . . . 14
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V)) |
33 | 32 | ancoms 461 |
. . . . . . . . . . . . 13
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V)) |
34 | | isuspgrop 26949 |
. . . . . . . . . . . . 13
⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (〈𝑉, ( I ↾ 𝑎)〉 ∈ USPGraph ↔ ( I ↾
𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) |
35 | 33, 34 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (〈𝑉, ( I ↾ 𝑎)〉 ∈ USPGraph ↔ ( I ↾
𝑎):dom ( I ↾ 𝑎)–1-1→{𝑝 ∈ (𝒫 𝑉 ∖ {∅}) ∣
(♯‘𝑝) ≤
2})) |
36 | 29, 35 | mpbird 259 |
. . . . . . . . . . 11
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 〈𝑉, ( I ↾ 𝑎)〉 ∈ USPGraph) |
37 | | fveqeq2 6682 |
. . . . . . . . . . . . 13
⊢ (𝑞 = 〈𝑉, ( I ↾ 𝑎)〉 → ((Vtx‘𝑞) = 𝑉 ↔ (Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉)) |
38 | | fveqeq2 6682 |
. . . . . . . . . . . . 13
⊢ (𝑞 = 〈𝑉, ( I ↾ 𝑎)〉 → ((Edg‘𝑞) = 𝑎 ↔ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎)) |
39 | 37, 38 | anbi12d 632 |
. . . . . . . . . . . 12
⊢ (𝑞 = 〈𝑉, ( I ↾ 𝑎)〉 → (((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎) ↔ ((Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉 ∧ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎))) |
40 | 39 | adantl 484 |
. . . . . . . . . . 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))) |
41 | | opvtxfv 26792 |
. . . . . . . . . . . . . 14
⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉) |
42 | 32, 41 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉) |
43 | | edgopval 26839 |
. . . . . . . . . . . . . . 15
⊢ ((𝑉 ∈ 𝑊 ∧ ( I ↾ 𝑎) ∈ V) → (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = ran ( I ↾ 𝑎)) |
44 | 32, 43 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = ran ( I ↾ 𝑎)) |
45 | | rnresi 5946 |
. . . . . . . . . . . . . 14
⊢ ran ( I
↾ 𝑎) = 𝑎 |
46 | 44, 45 | syl6eq 2875 |
. . . . . . . . . . . . 13
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎) |
47 | 42, 46 | jca 514 |
. . . . . . . . . . . 12
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ((Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉 ∧ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎)) |
48 | 47 | ancoms 461 |
. . . . . . . . . . 11
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ((Vtx‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑉 ∧ (Edg‘〈𝑉, ( I ↾ 𝑎)〉) = 𝑎)) |
49 | 36, 40, 48 | rspcedvd 3629 |
. . . . . . . . . 10
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)) |
50 | 9, 49 | jca 514 |
. . . . . . . . 9
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))) |
51 | 2 | eleq2i 2907 |
. . . . . . . . . 10
⊢
(〈𝑉, 𝑎〉 ∈ 𝐺 ↔ 〈𝑉, 𝑎〉 ∈ {〈𝑣, 𝑒〉 ∣ (𝑣 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒))}) |
52 | 30 | anim1i 616 |
. . . . . . . . . . . 12
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑎 ∈ V ∧ 𝑉 ∈ 𝑊)) |
53 | 52 | ancomd 464 |
. . . . . . . . . . 11
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V)) |
54 | | eqeq1 2828 |
. . . . . . . . . . . . . 14
⊢ (𝑣 = 𝑉 → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉)) |
55 | 54 | adantr 483 |
. . . . . . . . . . . . 13
⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (𝑣 = 𝑉 ↔ 𝑉 = 𝑉)) |
56 | | eqeq2 2836 |
. . . . . . . . . . . . . . 15
⊢ (𝑣 = 𝑉 → ((Vtx‘𝑞) = 𝑣 ↔ (Vtx‘𝑞) = 𝑉)) |
57 | | eqeq2 2836 |
. . . . . . . . . . . . . . 15
⊢ (𝑒 = 𝑎 → ((Edg‘𝑞) = 𝑒 ↔ (Edg‘𝑞) = 𝑎)) |
58 | 56, 57 | bi2anan9 637 |
. . . . . . . . . . . . . 14
⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))) |
59 | 58 | rexbidv 3300 |
. . . . . . . . . . . . 13
⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝑎) → (∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑣 ∧ (Edg‘𝑞) = 𝑒) ↔ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎))) |
60 | 55, 59 | 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)))) |
61 | 60 | opelopabga 5423 |
. . . . . . . . . . 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)))) |
62 | 53, 61 | 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)))) |
63 | 51, 62 | syl5bb 285 |
. . . . . . . . 9
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (〈𝑉, 𝑎〉 ∈ 𝐺 ↔ (𝑉 = 𝑉 ∧ ∃𝑞 ∈ USPGraph ((Vtx‘𝑞) = 𝑉 ∧ (Edg‘𝑞) = 𝑎)))) |
64 | 50, 63 | mpbird 259 |
. . . . . . . 8
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 〈𝑉, 𝑎〉 ∈ 𝐺) |
65 | | fveq2 6673 |
. . . . . . . . . 10
⊢ (𝑏 = 〈𝑉, 𝑎〉 → (2nd ‘𝑏) = (2nd
‘〈𝑉, 𝑎〉)) |
66 | 65 | eqeq2d 2835 |
. . . . . . . . 9
⊢ (𝑏 = 〈𝑉, 𝑎〉 → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘〈𝑉, 𝑎〉))) |
67 | 66 | adantl 484 |
. . . . . . . 8
⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) ∧ 𝑏 = 〈𝑉, 𝑎〉) → (𝑎 = (2nd ‘𝑏) ↔ 𝑎 = (2nd ‘〈𝑉, 𝑎〉))) |
68 | | op2ndg 7705 |
. . . . . . . . . . 11
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ V) → (2nd
‘〈𝑉, 𝑎〉) = 𝑎) |
69 | 68 | elvd 3503 |
. . . . . . . . . 10
⊢ (𝑉 ∈ 𝑊 → (2nd ‘〈𝑉, 𝑎〉) = 𝑎) |
70 | 69 | adantl 484 |
. . . . . . . . 9
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → (2nd ‘〈𝑉, 𝑎〉) = 𝑎) |
71 | 70 | eqcomd 2830 |
. . . . . . . 8
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → 𝑎 = (2nd ‘〈𝑉, 𝑎〉)) |
72 | 64, 67, 71 | rspcedvd 3629 |
. . . . . . 7
⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑉 ∈ 𝑊) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)) |
73 | 72 | ex 415 |
. . . . . 6
⊢ (𝑎 ⊆ (Pairs‘𝑉) → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))) |
74 | 8, 73 | sylbi 219 |
. . . . 5
⊢ (𝑎 ∈ 𝑃 → (𝑉 ∈ 𝑊 → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))) |
75 | 74 | impcom 410 |
. . . 4
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏)) |
76 | 1, 2, 3 | uspgrsprfv 44027 |
. . . . . . 7
⊢ (𝑏 ∈ 𝐺 → (𝐹‘𝑏) = (2nd ‘𝑏)) |
77 | 76 | adantl 484 |
. . . . . 6
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝐹‘𝑏) = (2nd ‘𝑏)) |
78 | 77 | eqeq2d 2835 |
. . . . 5
⊢ (((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) ∧ 𝑏 ∈ 𝐺) → (𝑎 = (𝐹‘𝑏) ↔ 𝑎 = (2nd ‘𝑏))) |
79 | 78 | rexbidva 3299 |
. . . 4
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → (∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏) ↔ ∃𝑏 ∈ 𝐺 𝑎 = (2nd ‘𝑏))) |
80 | 75, 79 | mpbird 259 |
. . 3
⊢ ((𝑉 ∈ 𝑊 ∧ 𝑎 ∈ 𝑃) → ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏)) |
81 | 80 | ralrimiva 3185 |
. 2
⊢ (𝑉 ∈ 𝑊 → ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏)) |
82 | | dffo3 6871 |
. 2
⊢ (𝐹:𝐺–onto→𝑃 ↔ (𝐹:𝐺⟶𝑃 ∧ ∀𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝐺 𝑎 = (𝐹‘𝑏))) |
83 | 5, 81, 82 | sylanbrc 585 |
1
⊢ (𝑉 ∈ 𝑊 → 𝐹:𝐺–onto→𝑃) |