| Step | Hyp | Ref
| Expression |
| 1 | | eldifi 4131 |
. . . . . 6
⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → 𝑧 ∈ 𝐴) |
| 2 | | frrlem13.8 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ∈ V) |
| 3 | 1, 2 | sylan2 593 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑆 ∈ V) |
| 4 | 3 | adantrr 717 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑆 ∈ V) |
| 5 | | inex1g 5319 |
. . . . 5
⊢ (𝑆 ∈ V → (𝑆 ∩ dom 𝐹) ∈ V) |
| 6 | | snex 5436 |
. . . . 5
⊢ {𝑧} ∈ V |
| 7 | | unexg 7763 |
. . . . 5
⊢ (((𝑆 ∩ dom 𝐹) ∈ V ∧ {𝑧} ∈ V) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V) |
| 8 | 5, 6, 7 | sylancl 586 |
. . . 4
⊢ (𝑆 ∈ V → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V) |
| 9 | 4, 8 | syl 17 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V) |
| 10 | | frrlem11.1 |
. . . . 5
⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
| 11 | | frrlem11.2 |
. . . . 5
⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
| 12 | | frrlem11.3 |
. . . . 5
⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
| 13 | | frrlem11.4 |
. . . . 5
⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) |
| 14 | 10, 11, 12, 13 | frrlem11 8321 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
| 15 | 14 | adantrr 717 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
| 16 | | inss1 4237 |
. . . . . 6
⊢ (𝑆 ∩ dom 𝐹) ⊆ 𝑆 |
| 17 | | frrlem13.9 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → 𝑆 ⊆ 𝐴) |
| 18 | 1, 17 | sylan2 593 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑆 ⊆ 𝐴) |
| 19 | 18 | adantrr 717 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑆 ⊆ 𝐴) |
| 20 | 16, 19 | sstrid 3995 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑆 ∩ dom 𝐹) ⊆ 𝐴) |
| 21 | 1 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝑧 ∈ 𝐴) |
| 22 | 21 | adantrr 717 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝑧 ∈ 𝐴) |
| 23 | 22 | snssd 4809 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → {𝑧} ⊆ 𝐴) |
| 24 | 20, 23 | unssd 4192 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴) |
| 25 | | elun 4153 |
. . . . . . . . 9
⊢ (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧})) |
| 26 | | elin 3967 |
. . . . . . . . . 10
⊢ (𝑤 ∈ (𝑆 ∩ dom 𝐹) ↔ (𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹)) |
| 27 | | velsn 4642 |
. . . . . . . . . 10
⊢ (𝑤 ∈ {𝑧} ↔ 𝑤 = 𝑧) |
| 28 | 26, 27 | orbi12i 915 |
. . . . . . . . 9
⊢ ((𝑤 ∈ (𝑆 ∩ dom 𝐹) ∨ 𝑤 ∈ {𝑧}) ↔ ((𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧)) |
| 29 | 25, 28 | bitri 275 |
. . . . . . . 8
⊢ (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ ((𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧)) |
| 30 | | frrlem12.7 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ∀𝑤 ∈ 𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆) |
| 31 | 1, 30 | sylan2 593 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤 ∈ 𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆) |
| 32 | 31 | adantrr 717 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∀𝑤 ∈ 𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆) |
| 33 | | rsp 3247 |
. . . . . . . . . . . 12
⊢
(∀𝑤 ∈
𝑆 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆 → (𝑤 ∈ 𝑆 → Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)) |
| 34 | 32, 33 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑤 ∈ 𝑆 → Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆)) |
| 35 | 10, 11 | frrlem8 8318 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹) |
| 36 | 34, 35 | anim12d1 610 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹) → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆 ∧ Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹))) |
| 37 | | ssin 4239 |
. . . . . . . . . 10
⊢
((Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑆 ∧ Pred(𝑅, 𝐴, 𝑤) ⊆ dom 𝐹) ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹)) |
| 38 | 36, 37 | imbitrdi 251 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))) |
| 39 | | frrlem12.6 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆) |
| 40 | 1, 39 | sylan2 593 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆) |
| 41 | 40 | adantrr 717 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ 𝑆) |
| 42 | | preddif 6350 |
. . . . . . . . . . . . . . . 16
⊢
Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) |
| 43 | 42 | eqeq1i 2742 |
. . . . . . . . . . . . . . 15
⊢
(Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ ↔ (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) = ∅) |
| 44 | | ssdif0 4366 |
. . . . . . . . . . . . . . 15
⊢
(Pred(𝑅, 𝐴, 𝑧) ⊆ Pred(𝑅, dom 𝐹, 𝑧) ↔ (Pred(𝑅, 𝐴, 𝑧) ∖ Pred(𝑅, dom 𝐹, 𝑧)) = ∅) |
| 45 | 43, 44 | sylbb2 238 |
. . . . . . . . . . . . . 14
⊢
(Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) ⊆ Pred(𝑅, dom 𝐹, 𝑧)) |
| 46 | | predss 6329 |
. . . . . . . . . . . . . 14
⊢
Pred(𝑅, dom 𝐹, 𝑧) ⊆ dom 𝐹 |
| 47 | 45, 46 | sstrdi 3996 |
. . . . . . . . . . . . 13
⊢
(Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅ → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |
| 48 | 47 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |
| 49 | 48 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ dom 𝐹) |
| 50 | 41, 49 | ssind 4241 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → Pred(𝑅, 𝐴, 𝑧) ⊆ (𝑆 ∩ dom 𝐹)) |
| 51 | | predeq3 6325 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑧)) |
| 52 | 51 | sseq1d 4015 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹) ↔ Pred(𝑅, 𝐴, 𝑧) ⊆ (𝑆 ∩ dom 𝐹))) |
| 53 | 50, 52 | syl5ibrcom 247 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑤 = 𝑧 → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))) |
| 54 | 38, 53 | jaod 860 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (((𝑤 ∈ 𝑆 ∧ 𝑤 ∈ dom 𝐹) ∨ 𝑤 = 𝑧) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))) |
| 55 | 29, 54 | biimtrid 242 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹))) |
| 56 | 55 | imp 406 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → Pred(𝑅, 𝐴, 𝑤) ⊆ (𝑆 ∩ dom 𝐹)) |
| 57 | | ssun1 4178 |
. . . . . 6
⊢ (𝑆 ∩ dom 𝐹) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) |
| 58 | 56, 57 | sstrdi 3996 |
. . . . 5
⊢ (((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
| 59 | 58 | ralrimiva 3146 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
| 60 | 24, 59 | jca 511 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))) |
| 61 | | frrlem12.5 |
. . . . . . 7
⊢ (𝜑 → 𝑅 Fr 𝐴) |
| 62 | 10, 11, 12, 13, 61, 39, 30 | frrlem12 8322 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) |
| 63 | 62 | 3expa 1119 |
. . . . 5
⊢ (((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) ∧ 𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) → (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) |
| 64 | 63 | ralrimiva 3146 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) |
| 65 | 64 | adantrr 717 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) |
| 66 | | fneq2 6660 |
. . . . . 6
⊢ (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝐶 Fn 𝑡 ↔ 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))) |
| 67 | | sseq1 4009 |
. . . . . . 7
⊢ (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (𝑡 ⊆ 𝐴 ↔ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴)) |
| 68 | | sseq2 4010 |
. . . . . . . 8
⊢ (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡 ↔ Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))) |
| 69 | 68 | raleqbi1dv 3338 |
. . . . . . 7
⊢ (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡 ↔ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧}))) |
| 70 | 67, 69 | anbi12d 632 |
. . . . . 6
⊢ (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → ((𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ↔ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})))) |
| 71 | | raleq 3323 |
. . . . . 6
⊢ (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → (∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
| 72 | 66, 70, 71 | 3anbi123d 1438 |
. . . . 5
⊢ (𝑡 = ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) → ((𝐶 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∧ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))) |
| 73 | 72 | spcegv 3597 |
. . . 4
⊢ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V → ((𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∧ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) → ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))) |
| 74 | 73 | imp 406 |
. . 3
⊢ ((((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∈ V ∧ (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ∧ (((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ⊆ 𝐴 ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})Pred(𝑅, 𝐴, 𝑤) ⊆ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) ∧ ∀𝑤 ∈ ((𝑆 ∩ dom 𝐹) ∪ {𝑧})(𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))) → ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
| 75 | 9, 15, 60, 65, 74 | syl13anc 1374 |
. 2
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
| 76 | 10, 11, 12 | frrlem9 8319 |
. . . . . 6
⊢ (𝜑 → Fun 𝐹) |
| 77 | | resfunexg 7235 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ 𝑆 ∈ V) → (𝐹 ↾ 𝑆) ∈ V) |
| 78 | 76, 4, 77 | syl2an2r 685 |
. . . . 5
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝐹 ↾ 𝑆) ∈ V) |
| 79 | | snex 5436 |
. . . . 5
⊢
{〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} ∈ V |
| 80 | | unexg 7763 |
. . . . 5
⊢ (((𝐹 ↾ 𝑆) ∈ V ∧ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} ∈ V) → ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) ∈ V) |
| 81 | 78, 79, 80 | sylancl 586 |
. . . 4
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) ∈ V) |
| 82 | 13, 81 | eqeltrid 2845 |
. . 3
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ V) |
| 83 | | fneq1 6659 |
. . . . . 6
⊢ (𝑐 = 𝐶 → (𝑐 Fn 𝑡 ↔ 𝐶 Fn 𝑡)) |
| 84 | | fveq1 6905 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → (𝑐‘𝑤) = (𝐶‘𝑤)) |
| 85 | | reseq1 5991 |
. . . . . . . . 9
⊢ (𝑐 = 𝐶 → (𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)) = (𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))) |
| 86 | 85 | oveq2d 7447 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))) |
| 87 | 84, 86 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑐 = 𝐶 → ((𝑐‘𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
| 88 | 87 | ralbidv 3178 |
. . . . . 6
⊢ (𝑐 = 𝐶 → (∀𝑤 ∈ 𝑡 (𝑐‘𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))) ↔ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
| 89 | 83, 88 | 3anbi13d 1440 |
. . . . 5
⊢ (𝑐 = 𝐶 → ((𝑐 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝑐‘𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ (𝐶 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))) |
| 90 | 89 | exbidv 1921 |
. . . 4
⊢ (𝑐 = 𝐶 → (∃𝑡(𝑐 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝑐‘𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤)))) ↔ ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))) |
| 91 | 10 | frrlem1 8311 |
. . . 4
⊢ 𝐵 = {𝑐 ∣ ∃𝑡(𝑐 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝑐‘𝑤) = (𝑤𝐺(𝑐 ↾ Pred(𝑅, 𝐴, 𝑤))))} |
| 92 | 90, 91 | elab2g 3680 |
. . 3
⊢ (𝐶 ∈ V → (𝐶 ∈ 𝐵 ↔ ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))) |
| 93 | 82, 92 | syl 17 |
. 2
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → (𝐶 ∈ 𝐵 ↔ ∃𝑡(𝐶 Fn 𝑡 ∧ (𝑡 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑡 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑡) ∧ ∀𝑤 ∈ 𝑡 (𝐶‘𝑤) = (𝑤𝐺(𝐶 ↾ Pred(𝑅, 𝐴, 𝑤)))))) |
| 94 | 75, 93 | mpbird 257 |
1
⊢ ((𝜑 ∧ (𝑧 ∈ (𝐴 ∖ dom 𝐹) ∧ Pred(𝑅, (𝐴 ∖ dom 𝐹), 𝑧) = ∅)) → 𝐶 ∈ 𝐵) |