Step | Hyp | Ref
| Expression |
1 | | reldom 8247 |
. . . 4
⊢ Rel
≼ |
2 | 1 | brrelex2i 5407 |
. . 3
⊢ (𝐵 ≼ 𝐴 → 𝐴 ∈ V) |
3 | 2 | adantl 475 |
. 2
⊢ ((∅
≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → 𝐴 ∈ V) |
4 | 1 | brrelex1i 5406 |
. . . 4
⊢ (𝐵 ≼ 𝐴 → 𝐵 ∈ V) |
5 | | 0sdomg 8377 |
. . . . 5
⊢ (𝐵 ∈ V → (∅
≺ 𝐵 ↔ 𝐵 ≠ ∅)) |
6 | | n0 4158 |
. . . . 5
⊢ (𝐵 ≠ ∅ ↔
∃𝑧 𝑧 ∈ 𝐵) |
7 | 5, 6 | syl6bb 279 |
. . . 4
⊢ (𝐵 ∈ V → (∅
≺ 𝐵 ↔
∃𝑧 𝑧 ∈ 𝐵)) |
8 | 4, 7 | syl 17 |
. . 3
⊢ (𝐵 ≼ 𝐴 → (∅ ≺ 𝐵 ↔ ∃𝑧 𝑧 ∈ 𝐵)) |
9 | 8 | biimpac 472 |
. 2
⊢ ((∅
≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑧 𝑧 ∈ 𝐵) |
10 | | brdomi 8252 |
. . 3
⊢ (𝐵 ≼ 𝐴 → ∃𝑔 𝑔:𝐵–1-1→𝐴) |
11 | 10 | adantl 475 |
. 2
⊢ ((∅
≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑔 𝑔:𝐵–1-1→𝐴) |
12 | | difexg 5045 |
. . . . . . . . . 10
⊢ (𝐴 ∈ V → (𝐴 ∖ ran 𝑔) ∈ V) |
13 | | snex 5140 |
. . . . . . . . . 10
⊢ {𝑧} ∈ V |
14 | | xpexg 7237 |
. . . . . . . . . 10
⊢ (((𝐴 ∖ ran 𝑔) ∈ V ∧ {𝑧} ∈ V) → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V) |
15 | 12, 13, 14 | sylancl 580 |
. . . . . . . . 9
⊢ (𝐴 ∈ V → ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V) |
16 | | vex 3400 |
. . . . . . . . . 10
⊢ 𝑔 ∈ V |
17 | 16 | cnvex 7392 |
. . . . . . . . 9
⊢ ◡𝑔 ∈ V |
18 | 15, 17 | jctil 515 |
. . . . . . . 8
⊢ (𝐴 ∈ V → (◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V)) |
19 | | unexb 7235 |
. . . . . . . 8
⊢ ((◡𝑔 ∈ V ∧ ((𝐴 ∖ ran 𝑔) × {𝑧}) ∈ V) ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V) |
20 | 18, 19 | sylib 210 |
. . . . . . 7
⊢ (𝐴 ∈ V → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V) |
21 | | df-f1 6140 |
. . . . . . . . . . . . 13
⊢ (𝑔:𝐵–1-1→𝐴 ↔ (𝑔:𝐵⟶𝐴 ∧ Fun ◡𝑔)) |
22 | 21 | simprbi 492 |
. . . . . . . . . . . 12
⊢ (𝑔:𝐵–1-1→𝐴 → Fun ◡𝑔) |
23 | | vex 3400 |
. . . . . . . . . . . . . 14
⊢ 𝑧 ∈ V |
24 | 23 | fconst 6341 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧} |
25 | | ffun 6294 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∖ ran 𝑔) × {𝑧}):(𝐴 ∖ ran 𝑔)⟶{𝑧} → Fun ((𝐴 ∖ ran 𝑔) × {𝑧})) |
26 | 24, 25 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ Fun
((𝐴 ∖ ran 𝑔) × {𝑧}) |
27 | 22, 26 | jctir 516 |
. . . . . . . . . . 11
⊢ (𝑔:𝐵–1-1→𝐴 → (Fun ◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧}))) |
28 | | df-rn 5366 |
. . . . . . . . . . . . . 14
⊢ ran 𝑔 = dom ◡𝑔 |
29 | 28 | eqcomi 2786 |
. . . . . . . . . . . . 13
⊢ dom ◡𝑔 = ran 𝑔 |
30 | 23 | snnz 4541 |
. . . . . . . . . . . . . 14
⊢ {𝑧} ≠ ∅ |
31 | | dmxp 5589 |
. . . . . . . . . . . . . 14
⊢ ({𝑧} ≠ ∅ → dom
((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔)) |
32 | 30, 31 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ dom
((𝐴 ∖ ran 𝑔) × {𝑧}) = (𝐴 ∖ ran 𝑔) |
33 | 29, 32 | ineq12i 4034 |
. . . . . . . . . . . 12
⊢ (dom
◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∩ (𝐴 ∖ ran 𝑔)) |
34 | | disjdif 4263 |
. . . . . . . . . . . 12
⊢ (ran
𝑔 ∩ (𝐴 ∖ ran 𝑔)) = ∅ |
35 | 33, 34 | eqtri 2801 |
. . . . . . . . . . 11
⊢ (dom
◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅ |
36 | | funun 6180 |
. . . . . . . . . . 11
⊢ (((Fun
◡𝑔 ∧ Fun ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ (dom ◡𝑔 ∩ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = ∅) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧}))) |
37 | 27, 35, 36 | sylancl 580 |
. . . . . . . . . 10
⊢ (𝑔:𝐵–1-1→𝐴 → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧}))) |
38 | 37 | adantl 475 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧}))) |
39 | | dmun 5576 |
. . . . . . . . . . . 12
⊢ dom
(◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) |
40 | 28 | uneq1i 3985 |
. . . . . . . . . . . 12
⊢ (ran
𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (dom ◡𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) |
41 | 32 | uneq2i 3986 |
. . . . . . . . . . . 12
⊢ (ran
𝑔 ∪ dom ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) |
42 | 39, 40, 41 | 3eqtr2i 2807 |
. . . . . . . . . . 11
⊢ dom
(◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) |
43 | | f1f 6351 |
. . . . . . . . . . . . 13
⊢ (𝑔:𝐵–1-1→𝐴 → 𝑔:𝐵⟶𝐴) |
44 | 43 | frnd 6298 |
. . . . . . . . . . . 12
⊢ (𝑔:𝐵–1-1→𝐴 → ran 𝑔 ⊆ 𝐴) |
45 | | undif 4272 |
. . . . . . . . . . . 12
⊢ (ran
𝑔 ⊆ 𝐴 ↔ (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴) |
46 | 44, 45 | sylib 210 |
. . . . . . . . . . 11
⊢ (𝑔:𝐵–1-1→𝐴 → (ran 𝑔 ∪ (𝐴 ∖ ran 𝑔)) = 𝐴) |
47 | 42, 46 | syl5eq 2825 |
. . . . . . . . . 10
⊢ (𝑔:𝐵–1-1→𝐴 → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴) |
48 | 47 | adantl 475 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴) |
49 | | df-fn 6138 |
. . . . . . . . 9
⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ↔ (Fun (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∧ dom (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐴)) |
50 | 38, 48, 49 | sylanbrc 578 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴) |
51 | | rnun 5795 |
. . . . . . . . 9
⊢ ran
(◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) |
52 | | dfdm4 5561 |
. . . . . . . . . . . 12
⊢ dom 𝑔 = ran ◡𝑔 |
53 | | f1dm 6355 |
. . . . . . . . . . . 12
⊢ (𝑔:𝐵–1-1→𝐴 → dom 𝑔 = 𝐵) |
54 | 52, 53 | syl5eqr 2827 |
. . . . . . . . . . 11
⊢ (𝑔:𝐵–1-1→𝐴 → ran ◡𝑔 = 𝐵) |
55 | 54 | uneq1d 3988 |
. . . . . . . . . 10
⊢ (𝑔:𝐵–1-1→𝐴 → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧}))) |
56 | | xpeq1 5369 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = (∅ × {𝑧})) |
57 | | 0xp 5447 |
. . . . . . . . . . . . . . . . 17
⊢ (∅
× {𝑧}) =
∅ |
58 | 56, 57 | syl6eq 2829 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅) |
59 | 58 | rneqd 5598 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ran ∅) |
60 | | rn0 5623 |
. . . . . . . . . . . . . . 15
⊢ ran
∅ = ∅ |
61 | 59, 60 | syl6eq 2829 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = ∅) |
62 | | 0ss 4197 |
. . . . . . . . . . . . . 14
⊢ ∅
⊆ 𝐵 |
63 | 61, 62 | syl6eqss 3873 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∖ ran 𝑔) = ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵) |
64 | 63 | a1d 25 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∖ ran 𝑔) = ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)) |
65 | | rnxp 5818 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧}) |
66 | 65 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) = {𝑧}) |
67 | | snssi 4570 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 ∈ 𝐵 → {𝑧} ⊆ 𝐵) |
68 | 67 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → {𝑧} ⊆ 𝐵) |
69 | 66, 68 | eqsstrd 3857 |
. . . . . . . . . . . . 13
⊢ (((𝐴 ∖ ran 𝑔) ≠ ∅ ∧ 𝑧 ∈ 𝐵) → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵) |
70 | 69 | ex 403 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∖ ran 𝑔) ≠ ∅ → (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵)) |
71 | 64, 70 | pm2.61ine 3052 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ 𝐵 → ran ((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵) |
72 | | ssequn2 4008 |
. . . . . . . . . . 11
⊢ (ran
((𝐴 ∖ ran 𝑔) × {𝑧}) ⊆ 𝐵 ↔ (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵) |
73 | 71, 72 | sylib 210 |
. . . . . . . . . 10
⊢ (𝑧 ∈ 𝐵 → (𝐵 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵) |
74 | 55, 73 | sylan9eqr 2835 |
. . . . . . . . 9
⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (ran ◡𝑔 ∪ ran ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵) |
75 | 51, 74 | syl5eq 2825 |
. . . . . . . 8
⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵) |
76 | | df-fo 6141 |
. . . . . . . 8
⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵 ↔ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) Fn 𝐴 ∧ ran (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) = 𝐵)) |
77 | 50, 75, 76 | sylanbrc 578 |
. . . . . . 7
⊢ ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵) |
78 | | foeq1 6362 |
. . . . . . . 8
⊢ (𝑓 = (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) → (𝑓:𝐴–onto→𝐵 ↔ (◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵)) |
79 | 78 | spcegv 3495 |
. . . . . . 7
⊢ ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})) ∈ V → ((◡𝑔 ∪ ((𝐴 ∖ ran 𝑔) × {𝑧})):𝐴–onto→𝐵 → ∃𝑓 𝑓:𝐴–onto→𝐵)) |
80 | 20, 77, 79 | syl2im 40 |
. . . . . 6
⊢ (𝐴 ∈ V → ((𝑧 ∈ 𝐵 ∧ 𝑔:𝐵–1-1→𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵)) |
81 | 80 | expdimp 446 |
. . . . 5
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵)) |
82 | 81 | exlimdv 1976 |
. . . 4
⊢ ((𝐴 ∈ V ∧ 𝑧 ∈ 𝐵) → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵)) |
83 | 82 | ex 403 |
. . 3
⊢ (𝐴 ∈ V → (𝑧 ∈ 𝐵 → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵))) |
84 | 83 | exlimdv 1976 |
. 2
⊢ (𝐴 ∈ V → (∃𝑧 𝑧 ∈ 𝐵 → (∃𝑔 𝑔:𝐵–1-1→𝐴 → ∃𝑓 𝑓:𝐴–onto→𝐵))) |
85 | 3, 9, 11, 84 | syl3c 66 |
1
⊢ ((∅
≺ 𝐵 ∧ 𝐵 ≼ 𝐴) → ∃𝑓 𝑓:𝐴–onto→𝐵) |