Step | Hyp | Ref
| Expression |
1 | | ensym 8005 |
. 2
⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐵 ∪ 𝐶) ≈ (𝐴 × 𝐴)) |
2 | | bren 7964 |
. . 3
⊢ ((𝐵 ∪ 𝐶) ≈ (𝐴 × 𝐴) ↔ ∃𝑓 𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴)) |
3 | | ssdif0 3942 |
. . . . . 6
⊢ (𝐴 ⊆ (((1st
↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ↔ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅) |
4 | | dmxpid 5345 |
. . . . . . . . . . . . . 14
⊢ dom
(𝐴 × 𝐴) = 𝐴 |
5 | | f1ofo 6144 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵 ∪ 𝐶)–onto→(𝐴 × 𝐴)) |
6 | | forn 6118 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓:(𝐵 ∪ 𝐶)–onto→(𝐴 × 𝐴) → ran 𝑓 = (𝐴 × 𝐴)) |
7 | 5, 6 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ran 𝑓 = (𝐴 × 𝐴)) |
8 | | vex 3203 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑓 ∈ V |
9 | 8 | rnex 7100 |
. . . . . . . . . . . . . . . 16
⊢ ran 𝑓 ∈ V |
10 | 7, 9 | syl6eqelr 2710 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 × 𝐴) ∈ V) |
11 | | dmexg 7097 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 × 𝐴) ∈ V → dom (𝐴 × 𝐴) ∈ V) |
12 | 10, 11 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → dom (𝐴 × 𝐴) ∈ V) |
13 | 4, 12 | syl5eqelr 2706 |
. . . . . . . . . . . . 13
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐴 ∈ V) |
14 | | imassrn 5477 |
. . . . . . . . . . . . . 14
⊢
(((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ ran ((1st ↾
(𝐴 × 𝐴)) ∘ 𝑓) |
15 | | f1stres 7190 |
. . . . . . . . . . . . . . . 16
⊢
(1st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴 |
16 | | f1of 6137 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴)) |
17 | | fco 6058 |
. . . . . . . . . . . . . . . 16
⊢
(((1st ↾ (𝐴 × 𝐴)):(𝐴 × 𝐴)⟶𝐴 ∧ 𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴)) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵 ∪ 𝐶)⟶𝐴) |
18 | 15, 16, 17 | sylancr 695 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵 ∪ 𝐶)⟶𝐴) |
19 | | frn 6053 |
. . . . . . . . . . . . . . 15
⊢
(((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵 ∪ 𝐶)⟶𝐴 → ran ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ⊆ 𝐴) |
20 | 18, 19 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ran ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) ⊆ 𝐴) |
21 | 14, 20 | syl5ss 3614 |
. . . . . . . . . . . . 13
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ⊆ 𝐴) |
22 | 13, 21 | ssexd 4805 |
. . . . . . . . . . . 12
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V) |
23 | 22 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V) |
24 | | simpr 477 |
. . . . . . . . . . 11
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
25 | | ssdomg 8001 |
. . . . . . . . . . 11
⊢
((((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V → (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) |
26 | 23, 24, 25 | sylc 65 |
. . . . . . . . . 10
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
27 | | domwdom 8479 |
. . . . . . . . . 10
⊢ (𝐴 ≼ (((1st
↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → 𝐴 ≼* (((1st
↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
28 | 26, 27 | syl 17 |
. . . . . . . . 9
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼* (((1st
↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
29 | | ffun 6048 |
. . . . . . . . . . . 12
⊢
(((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵 ∪ 𝐶)⟶𝐴 → Fun ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)) |
30 | 18, 29 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → Fun ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)) |
31 | | ssun1 3776 |
. . . . . . . . . . . 12
⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶) |
32 | | f1odm 6141 |
. . . . . . . . . . . . 13
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → dom 𝑓 = (𝐵 ∪ 𝐶)) |
33 | 8 | dmex 7099 |
. . . . . . . . . . . . 13
⊢ dom 𝑓 ∈ V |
34 | 32, 33 | syl6eqelr 2710 |
. . . . . . . . . . . 12
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐵 ∪ 𝐶) ∈ V) |
35 | | ssexg 4804 |
. . . . . . . . . . . 12
⊢ ((𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐵 ∈ V) |
36 | 31, 34, 35 | sylancr 695 |
. . . . . . . . . . 11
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐵 ∈ V) |
37 | | wdomima2g 8491 |
. . . . . . . . . . 11
⊢ ((Fun
((1st ↾ (𝐴
× 𝐴)) ∘ 𝑓) ∧ 𝐵 ∈ V ∧ (((1st ↾
(𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∈ V) → (((1st ↾
(𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵) |
38 | 30, 36, 22, 37 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵) |
39 | 38 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵) |
40 | | wdomtr 8480 |
. . . . . . . . 9
⊢ ((𝐴 ≼*
(((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ∧ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) ≼* 𝐵) → 𝐴 ≼* 𝐵) |
41 | 28, 39, 40 | syl2anc 693 |
. . . . . . . 8
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝐴 ≼* 𝐵) |
42 | 41 | orcd 407 |
. . . . . . 7
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) |
43 | 42 | ex 450 |
. . . . . 6
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ⊆ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))) |
44 | 3, 43 | syl5bir 233 |
. . . . 5
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) = ∅ → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))) |
45 | | n0 3931 |
. . . . . 6
⊢ ((𝐴 ∖ (((1st
↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) |
46 | | ssun2 3777 |
. . . . . . . . . . . . 13
⊢ 𝐶 ⊆ (𝐵 ∪ 𝐶) |
47 | | ssexg 4804 |
. . . . . . . . . . . . 13
⊢ ((𝐶 ⊆ (𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) → 𝐶 ∈ V) |
48 | 46, 34, 47 | sylancr 695 |
. . . . . . . . . . . 12
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝐶 ∈ V) |
49 | 48 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐶 ∈ V) |
50 | | f1ofn 6138 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → 𝑓 Fn (𝐵 ∪ 𝐶)) |
51 | | elpreima 6337 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 Fn (𝐵 ∪ 𝐶) → (𝑦 ∈ (◡𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵 ∪ 𝐶) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)))) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑦 ∈ (◡𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵 ∪ 𝐶) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)))) |
53 | 52 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (◡𝑓 “ ({𝑥} × 𝐴)) ↔ (𝑦 ∈ (𝐵 ∪ 𝐶) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)))) |
54 | | elun 3753 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶)) |
55 | | df-or 385 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (¬ 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
56 | 54, 55 | bitri 264 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (¬ 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶)) |
57 | | eldifn 3733 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → ¬ 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
58 | 57 | ad2antlr 763 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) → ¬ 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
59 | 16 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴)) |
60 | | simprr 796 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵) |
61 | 31, 60 | sseldi 3601 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ (𝐵 ∪ 𝐶)) |
62 | | fvco3 6275 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓:(𝐵 ∪ 𝐶)⟶(𝐴 × 𝐴) ∧ 𝑦 ∈ (𝐵 ∪ 𝐶)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = ((1st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦))) |
63 | 59, 61, 62 | syl2anc 693 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = ((1st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦))) |
64 | | eldifi 3732 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → 𝑥 ∈ 𝐴) |
65 | 64 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝑥 ∈ 𝐴) |
66 | 65 | snssd 4340 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → {𝑥} ⊆ 𝐴) |
67 | | xpss1 5228 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ({𝑥} ⊆ 𝐴 → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴)) |
68 | 66, 67 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴)) |
69 | 68 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴)) |
70 | | simprl 794 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) |
71 | 69, 70 | sseldd 3604 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (𝑓‘𝑦) ∈ (𝐴 × 𝐴)) |
72 | | fvres 6207 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓‘𝑦) ∈ (𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) = (1st ‘(𝑓‘𝑦))) |
73 | 71, 72 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) = (1st ‘(𝑓‘𝑦))) |
74 | | xp1st 7198 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) → (1st ‘(𝑓‘𝑦)) ∈ {𝑥}) |
75 | 70, 74 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (1st ‘(𝑓‘𝑦)) ∈ {𝑥}) |
76 | 73, 75 | eqeltrd 2701 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) ∈ {𝑥}) |
77 | | elsni 4194 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((1st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) ∈ {𝑥} → ((1st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) = 𝑥) |
78 | 76, 77 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ((1st ↾ (𝐴 × 𝐴))‘(𝑓‘𝑦)) = 𝑥) |
79 | 63, 78 | eqtrd 2656 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) = 𝑥) |
80 | | ffn 6045 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓):(𝐵 ∪ 𝐶)⟶𝐴 → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵 ∪ 𝐶)) |
81 | 18, 80 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵 ∪ 𝐶)) |
82 | 81 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → ((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵 ∪ 𝐶)) |
83 | 31 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝐵 ⊆ (𝐵 ∪ 𝐶)) |
84 | | fnfvima 6496 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) Fn (𝐵 ∪ 𝐶) ∧ 𝐵 ⊆ (𝐵 ∪ 𝐶) ∧ 𝑦 ∈ 𝐵) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
85 | 82, 83, 60, 84 | syl3anc 1326 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓)‘𝑦) ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
86 | 79, 85 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) |
87 | 86 | expr 643 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) → (𝑦 ∈ 𝐵 → 𝑥 ∈ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) |
88 | 58, 87 | mtod 189 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) → ¬ 𝑦 ∈ 𝐵) |
89 | 88 | ex 450 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) → ¬ 𝑦 ∈ 𝐵)) |
90 | 89 | imim1d 82 |
. . . . . . . . . . . . . . 15
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((¬ 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶) → ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) → 𝑦 ∈ 𝐶))) |
91 | 56, 90 | syl5bi 232 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (𝐵 ∪ 𝐶) → ((𝑓‘𝑦) ∈ ({𝑥} × 𝐴) → 𝑦 ∈ 𝐶))) |
92 | 91 | impd 447 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((𝑦 ∈ (𝐵 ∪ 𝐶) ∧ (𝑓‘𝑦) ∈ ({𝑥} × 𝐴)) → 𝑦 ∈ 𝐶)) |
93 | 53, 92 | sylbid 230 |
. . . . . . . . . . . 12
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝑦 ∈ (◡𝑓 “ ({𝑥} × 𝐴)) → 𝑦 ∈ 𝐶)) |
94 | 93 | ssrdv 3609 |
. . . . . . . . . . 11
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (◡𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶) |
95 | | ssdomg 8001 |
. . . . . . . . . . 11
⊢ (𝐶 ∈ V → ((◡𝑓 “ ({𝑥} × 𝐴)) ⊆ 𝐶 → (◡𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶)) |
96 | 49, 94, 95 | sylc 65 |
. . . . . . . . . 10
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (◡𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶) |
97 | | f1ocnv 6149 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ◡𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵 ∪ 𝐶)) |
98 | | f1of1 6136 |
. . . . . . . . . . . . . . 15
⊢ (◡𝑓:(𝐴 × 𝐴)–1-1-onto→(𝐵 ∪ 𝐶) → ◡𝑓:(𝐴 × 𝐴)–1-1→(𝐵 ∪ 𝐶)) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ◡𝑓:(𝐴 × 𝐴)–1-1→(𝐵 ∪ 𝐶)) |
100 | 99 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ◡𝑓:(𝐴 × 𝐴)–1-1→(𝐵 ∪ 𝐶)) |
101 | 34 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐵 ∪ 𝐶) ∈ V) |
102 | | snex 4908 |
. . . . . . . . . . . . . 14
⊢ {𝑥} ∈ V |
103 | 13 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ∈ V) |
104 | | xpexg 6960 |
. . . . . . . . . . . . . 14
⊢ (({𝑥} ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ∈ V) |
105 | 102, 103,
104 | sylancr 695 |
. . . . . . . . . . . . 13
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ∈ V) |
106 | | f1imaen2g 8017 |
. . . . . . . . . . . . 13
⊢ (((◡𝑓:(𝐴 × 𝐴)–1-1→(𝐵 ∪ 𝐶) ∧ (𝐵 ∪ 𝐶) ∈ V) ∧ (({𝑥} × 𝐴) ⊆ (𝐴 × 𝐴) ∧ ({𝑥} × 𝐴) ∈ V)) → (◡𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴)) |
107 | 100, 101,
68, 105, 106 | syl22anc 1327 |
. . . . . . . . . . . 12
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (◡𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴)) |
108 | | vex 3203 |
. . . . . . . . . . . . 13
⊢ 𝑥 ∈ V |
109 | | xpsnen2g 8053 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ V ∧ 𝐴 ∈ V) → ({𝑥} × 𝐴) ≈ 𝐴) |
110 | 108, 103,
109 | sylancr 695 |
. . . . . . . . . . . 12
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ({𝑥} × 𝐴) ≈ 𝐴) |
111 | | entr 8008 |
. . . . . . . . . . . 12
⊢ (((◡𝑓 “ ({𝑥} × 𝐴)) ≈ ({𝑥} × 𝐴) ∧ ({𝑥} × 𝐴) ≈ 𝐴) → (◡𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴) |
112 | 107, 110,
111 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (◡𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴) |
113 | | domen1 8102 |
. . . . . . . . . . 11
⊢ ((◡𝑓 “ ({𝑥} × 𝐴)) ≈ 𝐴 → ((◡𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶 ↔ 𝐴 ≼ 𝐶)) |
114 | 112, 113 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → ((◡𝑓 “ ({𝑥} × 𝐴)) ≼ 𝐶 ↔ 𝐴 ≼ 𝐶)) |
115 | 96, 114 | mpbid 222 |
. . . . . . . . 9
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → 𝐴 ≼ 𝐶) |
116 | 115 | olcd 408 |
. . . . . . . 8
⊢ ((𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) ∧ 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵))) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) |
117 | 116 | ex 450 |
. . . . . . 7
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))) |
118 | 117 | exlimdv 1861 |
. . . . . 6
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (∃𝑥 𝑥 ∈ (𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))) |
119 | 45, 118 | syl5bi 232 |
. . . . 5
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → ((𝐴 ∖ (((1st ↾ (𝐴 × 𝐴)) ∘ 𝑓) “ 𝐵)) ≠ ∅ → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶))) |
120 | 44, 119 | pm2.61dne 2880 |
. . . 4
⊢ (𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) |
121 | 120 | exlimiv 1858 |
. . 3
⊢
(∃𝑓 𝑓:(𝐵 ∪ 𝐶)–1-1-onto→(𝐴 × 𝐴) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) |
122 | 2, 121 | sylbi 207 |
. 2
⊢ ((𝐵 ∪ 𝐶) ≈ (𝐴 × 𝐴) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) |
123 | 1, 122 | syl 17 |
1
⊢ ((𝐴 × 𝐴) ≈ (𝐵 ∪ 𝐶) → (𝐴 ≼* 𝐵 ∨ 𝐴 ≼ 𝐶)) |