Step | Hyp | Ref
| Expression |
1 | | fpwwe2.1 |
. . . . . . . . . . 11
⊢ 𝑊 = {〈𝑥, 𝑟〉 ∣ ((𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥)) ∧ (𝑟 We 𝑥 ∧ ∀𝑦 ∈ 𝑥 [(◡𝑟 “ {𝑦}) / 𝑢](𝑢𝐹(𝑟 ∩ (𝑢 × 𝑢))) = 𝑦))} |
2 | | fpwwe2.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ V) |
3 | | fpwwe2.3 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) |
4 | | fpwwe2.4 |
. . . . . . . . . . 11
⊢ 𝑋 = ∪
dom 𝑊 |
5 | 1, 2, 3, 4 | fpwwe2lem11 9462 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋)) |
6 | | ffun 6048 |
. . . . . . . . . 10
⊢ (𝑊:dom 𝑊⟶𝒫 (𝑋 × 𝑋) → Fun 𝑊) |
7 | 5, 6 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → Fun 𝑊) |
8 | | funbrfv2b 6240 |
. . . . . . . . 9
⊢ (Fun
𝑊 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊‘𝑌) = 𝑅))) |
9 | 7, 8 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (𝑌𝑊𝑅 ↔ (𝑌 ∈ dom 𝑊 ∧ (𝑊‘𝑌) = 𝑅))) |
10 | 9 | simprbda 653 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌𝑊𝑅) → 𝑌 ∈ dom 𝑊) |
11 | 10 | adantrr 753 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ∈ dom 𝑊) |
12 | | elssuni 4467 |
. . . . . . 7
⊢ (𝑌 ∈ dom 𝑊 → 𝑌 ⊆ ∪ dom
𝑊) |
13 | 12, 4 | syl6sseqr 3652 |
. . . . . 6
⊢ (𝑌 ∈ dom 𝑊 → 𝑌 ⊆ 𝑋) |
14 | 11, 13 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 ⊆ 𝑋) |
15 | | simpl 473 |
. . . . . . 7
⊢ ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋 ⊆ 𝑌) |
16 | 15 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) → 𝑋 ⊆ 𝑌)) |
17 | | simplrr 801 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑌𝐹𝑅) ∈ 𝑌) |
18 | 2 | adantr 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝐴 ∈ V) |
19 | 18 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝐴 ∈ V) |
20 | 1, 2, 3, 4 | fpwwe2lem12 9463 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑋 ∈ dom 𝑊) |
21 | | funfvbrb 6330 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (Fun
𝑊 → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋))) |
22 | 7, 21 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝑋 ∈ dom 𝑊 ↔ 𝑋𝑊(𝑊‘𝑋))) |
23 | 20, 22 | mpbid 222 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑋𝑊(𝑊‘𝑋)) |
24 | 1, 2 | fpwwe2lem2 9454 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑋𝑊(𝑊‘𝑋) ↔ ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)))) |
25 | 23, 24 | mpbid 222 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))) |
26 | 25 | ad2antrr 762 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) ∧ ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦))) |
27 | 26 | simpld 475 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ⊆ 𝐴 ∧ (𝑊‘𝑋) ⊆ (𝑋 × 𝑋))) |
28 | 27 | simpld 475 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ⊆ 𝐴) |
29 | 19, 28 | ssexd 4805 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ∈ V) |
30 | | difexg 4808 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈ V → (𝑋 ∖ 𝑌) ∈ V) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) ∈ V) |
32 | 26 | simprd 479 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑊‘𝑋) We 𝑋 ∧ ∀𝑦 ∈ 𝑋 [(◡(𝑊‘𝑋) “ {𝑦}) / 𝑢](𝑢𝐹((𝑊‘𝑋) ∩ (𝑢 × 𝑢))) = 𝑦)) |
33 | 32 | simpld 475 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) We 𝑋) |
34 | | wefr 5104 |
. . . . . . . . . . . . 13
⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Fr 𝑋) |
35 | 33, 34 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) Fr 𝑋) |
36 | | difssd 3738 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) ⊆ 𝑋) |
37 | | fri 5076 |
. . . . . . . . . . . . 13
⊢ ((((𝑋 ∖ 𝑌) ∈ V ∧ (𝑊‘𝑋) Fr 𝑋) ∧ ((𝑋 ∖ 𝑌) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑌) ≠ ∅)) → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧) |
38 | 37 | expr 643 |
. . . . . . . . . . . 12
⊢ ((((𝑋 ∖ 𝑌) ∈ V ∧ (𝑊‘𝑋) Fr 𝑋) ∧ (𝑋 ∖ 𝑌) ⊆ 𝑋) → ((𝑋 ∖ 𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)) |
39 | 31, 35, 36, 38 | syl21anc 1325 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∖ 𝑌) ≠ ∅ → ∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧)) |
40 | | ssdif0 3942 |
. . . . . . . . . . . . . . 15
⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅) |
41 | | indif1 3871 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) |
42 | 41 | eqeq1i 2627 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ∖ 𝑌) = ∅) |
43 | | disj 4017 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧})) |
44 | | vex 3203 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑧 ∈ V |
45 | | vex 3203 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑤 ∈ V |
46 | 45 | eliniseg 5494 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 ∈ V → (𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ 𝑤(𝑊‘𝑋)𝑧)) |
47 | 44, 46 | ax-mp 5 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ 𝑤(𝑊‘𝑋)𝑧) |
48 | 47 | notbii 310 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ ¬ 𝑤(𝑊‘𝑋)𝑧) |
49 | 48 | ralbii 2980 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑤 ∈
(𝑋 ∖ 𝑌) ¬ 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}) ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧) |
50 | 43, 49 | bitri 264 |
. . . . . . . . . . . . . . 15
⊢ (((𝑋 ∖ 𝑌) ∩ (◡(𝑊‘𝑋) “ {𝑧})) = ∅ ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧) |
51 | 40, 42, 50 | 3bitr2i 288 |
. . . . . . . . . . . . . 14
⊢ ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ ∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧) |
52 | | cnvimass 5485 |
. . . . . . . . . . . . . . . . 17
⊢ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ dom (𝑊‘𝑋) |
53 | 27 | simprd 479 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑊‘𝑋) ⊆ (𝑋 × 𝑋)) |
54 | | dmss 5323 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑊‘𝑋) ⊆ (𝑋 × 𝑋) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋)) |
55 | 53, 54 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊‘𝑋) ⊆ dom (𝑋 × 𝑋)) |
56 | | dmxpid 5345 |
. . . . . . . . . . . . . . . . . 18
⊢ dom
(𝑋 × 𝑋) = 𝑋 |
57 | 55, 56 | syl6sseq 3651 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → dom (𝑊‘𝑋) ⊆ 𝑋) |
58 | 52, 57 | syl5ss 3614 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑋) |
59 | | sseqin2 3817 |
. . . . . . . . . . . . . . . 16
⊢ ((◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑋 ↔ (𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) = (◡(𝑊‘𝑋) “ {𝑧})) |
60 | 58, 59 | sylib 208 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) = (◡(𝑊‘𝑋) “ {𝑧})) |
61 | 60 | sseq1d 3632 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∩ (◡(𝑊‘𝑋) “ {𝑧})) ⊆ 𝑌 ↔ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) |
62 | 51, 61 | syl5bbr 274 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 ↔ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) |
63 | 62 | rexbidv 3052 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 ↔ ∃𝑧 ∈ (𝑋 ∖ 𝑌)(◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) |
64 | | eldifn 3733 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 ∈ (𝑋 ∖ 𝑌) → ¬ 𝑧 ∈ 𝑌) |
65 | 64 | ad2antrl 764 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ 𝑧 ∈ 𝑌) |
66 | | eleq1 2689 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝑌 ↔ 𝑧 ∈ 𝑌)) |
67 | 66 | notbid 308 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑤 = 𝑧 → (¬ 𝑤 ∈ 𝑌 ↔ ¬ 𝑧 ∈ 𝑌)) |
68 | 65, 67 | syl5ibrcom 237 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 = 𝑧 → ¬ 𝑤 ∈ 𝑌)) |
69 | 68 | con2d 129 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 ∈ 𝑌 → ¬ 𝑤 = 𝑧)) |
70 | 69 | imp 445 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑤 = 𝑧) |
71 | 65 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑧 ∈ 𝑌) |
72 | | simprr 796 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌))) |
73 | 72 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌))) |
74 | 73 | breqd 4664 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 ↔ 𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤)) |
75 | | eldifi 3732 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑧 ∈ (𝑋 ∖ 𝑌) → 𝑧 ∈ 𝑋) |
76 | 75 | ad2antrl 764 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑧 ∈ 𝑋) |
77 | 76 | adantr 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑧 ∈ 𝑋) |
78 | | simpr 477 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑌) |
79 | | brxp 5147 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧(𝑋 × 𝑌)𝑤 ↔ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌)) |
80 | 77, 78, 79 | sylanbrc 698 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑧(𝑋 × 𝑌)𝑤) |
81 | | brin 4704 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ (𝑧(𝑊‘𝑋)𝑤 ∧ 𝑧(𝑋 × 𝑌)𝑤)) |
82 | 81 | rbaib 947 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧(𝑋 × 𝑌)𝑤 → (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤)) |
83 | 80, 82 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧((𝑊‘𝑋) ∩ (𝑋 × 𝑌))𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤)) |
84 | 74, 83 | bitrd 268 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 ↔ 𝑧(𝑊‘𝑋)𝑤)) |
85 | 1, 2 | fpwwe2lem2 9454 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → (𝑌𝑊𝑅 ↔ ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦)))) |
86 | 85 | biimpa 501 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑌𝑊𝑅) → ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) |
87 | 86 | adantrr 753 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌)) ∧ (𝑅 We 𝑌 ∧ ∀𝑦 ∈ 𝑌 [(◡𝑅 “ {𝑦}) / 𝑢](𝑢𝐹(𝑅 ∩ (𝑢 × 𝑢))) = 𝑦))) |
88 | 87 | simpld 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 ⊆ 𝐴 ∧ 𝑅 ⊆ (𝑌 × 𝑌))) |
89 | 88 | simprd 479 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌)) |
90 | 89 | ad3antrrr 766 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑅 ⊆ (𝑌 × 𝑌)) |
91 | 90 | ssbrd 4696 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 → 𝑧(𝑌 × 𝑌)𝑤)) |
92 | | brxp 5147 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧(𝑌 × 𝑌)𝑤 ↔ (𝑧 ∈ 𝑌 ∧ 𝑤 ∈ 𝑌)) |
93 | 92 | simplbi 476 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧(𝑌 × 𝑌)𝑤 → 𝑧 ∈ 𝑌) |
94 | 91, 93 | syl6 35 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧𝑅𝑤 → 𝑧 ∈ 𝑌)) |
95 | 84, 94 | sylbird 250 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑧(𝑊‘𝑋)𝑤 → 𝑧 ∈ 𝑌)) |
96 | 71, 95 | mtod 189 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → ¬ 𝑧(𝑊‘𝑋)𝑤) |
97 | 33 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑊‘𝑋) We 𝑋) |
98 | | weso 5105 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑊‘𝑋) We 𝑋 → (𝑊‘𝑋) Or 𝑋) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑊‘𝑋) Or 𝑋) |
100 | 14 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ 𝑋) |
101 | 100 | sselda 3603 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ 𝑋) |
102 | | sotric 5061 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑊‘𝑋) Or 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑤(𝑊‘𝑋)𝑧 ↔ ¬ (𝑤 = 𝑧 ∨ 𝑧(𝑊‘𝑋)𝑤))) |
103 | | ioran 511 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
(𝑤 = 𝑧 ∨ 𝑧(𝑊‘𝑋)𝑤) ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤)) |
104 | 102, 103 | syl6bb 276 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑊‘𝑋) Or 𝑋 ∧ (𝑤 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋)) → (𝑤(𝑊‘𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤))) |
105 | 99, 101, 77, 104 | syl12anc 1324 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → (𝑤(𝑊‘𝑋)𝑧 ↔ (¬ 𝑤 = 𝑧 ∧ ¬ 𝑧(𝑊‘𝑋)𝑤))) |
106 | 70, 96, 105 | mpbir2and 957 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤(𝑊‘𝑋)𝑧) |
107 | 106, 47 | sylibr 224 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑤 ∈ 𝑌) → 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧})) |
108 | 107 | ex 450 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑤 ∈ 𝑌 → 𝑤 ∈ (◡(𝑊‘𝑋) “ {𝑧}))) |
109 | 108 | ssrdv 3609 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 ⊆ (◡(𝑊‘𝑋) “ {𝑧})) |
110 | | simprr 796 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌) |
111 | 109, 110 | eqssd 3620 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑌 = (◡(𝑊‘𝑋) “ {𝑧})) |
112 | | in32 3825 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) |
113 | | simplrr 801 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌))) |
114 | 113 | ineq1d 3813 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌))) |
115 | 89 | ad2antrr 762 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 ⊆ (𝑌 × 𝑌)) |
116 | | df-ss 3588 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑅 ⊆ (𝑌 × 𝑌) ↔ (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅) |
117 | 115, 116 | sylib 208 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑅 ∩ (𝑌 × 𝑌)) = 𝑅) |
118 | 114, 117 | eqtr3d 2658 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊‘𝑋) ∩ (𝑋 × 𝑌)) ∩ (𝑌 × 𝑌)) = 𝑅) |
119 | | inss2 3834 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑌 × 𝑌) |
120 | | xpss1 5228 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑌 ⊆ 𝑋 → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌)) |
121 | 100, 120 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) ⊆ (𝑋 × 𝑌)) |
122 | 119, 121 | syl5ss 3614 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌)) |
123 | | df-ss 3588 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ⊆ (𝑋 × 𝑌) ↔ (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌))) |
124 | 122, 123 | sylib 208 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) ∩ (𝑋 × 𝑌)) = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌))) |
125 | 112, 118,
124 | 3eqtr3a 2680 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ (𝑌 × 𝑌))) |
126 | 111 | sqxpeqd 5141 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌 × 𝑌) = ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧}))) |
127 | 126 | ineq2d 3814 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((𝑊‘𝑋) ∩ (𝑌 × 𝑌)) = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))) |
128 | 125, 127 | eqtrd 2656 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑅 = ((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))) |
129 | 111, 128 | oveq12d 6668 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧}))))) |
130 | 19 | adantr 481 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝐴 ∈ V) |
131 | 23 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊(𝑊‘𝑋)) |
132 | 131 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → 𝑋𝑊(𝑊‘𝑋)) |
133 | 1, 130, 132 | fpwwe2lem3 9455 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) ∧ 𝑧 ∈ 𝑋) → ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))) = 𝑧) |
134 | 76, 133 | mpdan 702 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ((◡(𝑊‘𝑋) “ {𝑧})𝐹((𝑊‘𝑋) ∩ ((◡(𝑊‘𝑋) “ {𝑧}) × (◡(𝑊‘𝑋) “ {𝑧})))) = 𝑧) |
135 | 129, 134 | eqtrd 2656 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → (𝑌𝐹𝑅) = 𝑧) |
136 | 135, 65 | eqneltrd 2720 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) ∧ (𝑧 ∈ (𝑋 ∖ 𝑌) ∧ (◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌)) → ¬ (𝑌𝐹𝑅) ∈ 𝑌) |
137 | 136 | rexlimdvaa 3032 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)(◡(𝑊‘𝑋) “ {𝑧}) ⊆ 𝑌 → ¬ (𝑌𝐹𝑅) ∈ 𝑌)) |
138 | 63, 137 | sylbid 230 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (∃𝑧 ∈ (𝑋 ∖ 𝑌)∀𝑤 ∈ (𝑋 ∖ 𝑌) ¬ 𝑤(𝑊‘𝑋)𝑧 → ¬ (𝑌𝐹𝑅) ∈ 𝑌)) |
139 | 39, 138 | syld 47 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑋 ∖ 𝑌) ≠ ∅ → ¬ (𝑌𝐹𝑅) ∈ 𝑌)) |
140 | 139 | necon4ad 2813 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → ((𝑌𝐹𝑅) ∈ 𝑌 → (𝑋 ∖ 𝑌) = ∅)) |
141 | 17, 140 | mpd 15 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → (𝑋 ∖ 𝑌) = ∅) |
142 | | ssdif0 3942 |
. . . . . . . 8
⊢ (𝑋 ⊆ 𝑌 ↔ (𝑋 ∖ 𝑌) = ∅) |
143 | 141, 142 | sylibr 224 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌)))) → 𝑋 ⊆ 𝑌) |
144 | 143 | ex 450 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌))) → 𝑋 ⊆ 𝑌)) |
145 | 3 | adantlr 751 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) ∧ (𝑥 ⊆ 𝐴 ∧ 𝑟 ⊆ (𝑥 × 𝑥) ∧ 𝑟 We 𝑥)) → (𝑥𝐹𝑟) ∈ 𝐴) |
146 | | simprl 794 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌𝑊𝑅) |
147 | 1, 18, 145, 131, 146 | fpwwe2lem10 9461 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → ((𝑋 ⊆ 𝑌 ∧ (𝑊‘𝑋) = (𝑅 ∩ (𝑌 × 𝑋))) ∨ (𝑌 ⊆ 𝑋 ∧ 𝑅 = ((𝑊‘𝑋) ∩ (𝑋 × 𝑌))))) |
148 | 16, 144, 147 | mpjaod 396 |
. . . . 5
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋 ⊆ 𝑌) |
149 | 14, 148 | eqssd 3620 |
. . . 4
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑌 = 𝑋) |
150 | 7 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → Fun 𝑊) |
151 | 149, 146 | eqbrtrrd 4677 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑋𝑊𝑅) |
152 | | funbrfv 6234 |
. . . . . 6
⊢ (Fun
𝑊 → (𝑋𝑊𝑅 → (𝑊‘𝑋) = 𝑅)) |
153 | 150, 151,
152 | sylc 65 |
. . . . 5
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑊‘𝑋) = 𝑅) |
154 | 153 | eqcomd 2628 |
. . . 4
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → 𝑅 = (𝑊‘𝑋)) |
155 | 149, 154 | jca 554 |
. . 3
⊢ ((𝜑 ∧ (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌)) → (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋))) |
156 | 155 | ex 450 |
. 2
⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) → (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)))) |
157 | 1, 2, 3, 4 | fpwwe2lem13 9464 |
. . . 4
⊢ (𝜑 → (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋) |
158 | 23, 157 | jca 554 |
. . 3
⊢ (𝜑 → (𝑋𝑊(𝑊‘𝑋) ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
159 | | breq12 4658 |
. . . 4
⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝑊𝑅 ↔ 𝑋𝑊(𝑊‘𝑋))) |
160 | | oveq12 6659 |
. . . . 5
⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝐹𝑅) = (𝑋𝐹(𝑊‘𝑋))) |
161 | | simpl 473 |
. . . . 5
⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → 𝑌 = 𝑋) |
162 | 160, 161 | eleq12d 2695 |
. . . 4
⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → ((𝑌𝐹𝑅) ∈ 𝑌 ↔ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋)) |
163 | 159, 162 | anbi12d 747 |
. . 3
⊢ ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑋𝑊(𝑊‘𝑋) ∧ (𝑋𝐹(𝑊‘𝑋)) ∈ 𝑋))) |
164 | 158, 163 | syl5ibrcom 237 |
. 2
⊢ (𝜑 → ((𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)) → (𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌))) |
165 | 156, 164 | impbid 202 |
1
⊢ (𝜑 → ((𝑌𝑊𝑅 ∧ (𝑌𝐹𝑅) ∈ 𝑌) ↔ (𝑌 = 𝑋 ∧ 𝑅 = (𝑊‘𝑋)))) |