| Step | Hyp | Ref
| Expression |
| 1 | | wfrlem6OLD.1 |
. . . . . . . 8
⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
| 2 | | dfwrecsOLD 8338 |
. . . . . . . 8
⊢
wrecs(𝑅, 𝐴, 𝐺) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
| 3 | 1, 2 | eqtri 2765 |
. . . . . . 7
⊢ 𝐹 = ∪
{𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
| 4 | 3 | dmeqi 5915 |
. . . . . 6
⊢ dom 𝐹 = dom ∪ {𝑓
∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
| 5 | | dmuni 5925 |
. . . . . 6
⊢ dom ∪ {𝑓
∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = ∪
𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔 |
| 6 | 4, 5 | eqtri 2765 |
. . . . 5
⊢ dom 𝐹 = ∪ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔 |
| 7 | 6 | eleq2i 2833 |
. . . 4
⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ ∪
𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔) |
| 8 | | eliun 4995 |
. . . 4
⊢ (𝑋 ∈ ∪ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔 ↔ ∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}𝑋 ∈ dom 𝑔) |
| 9 | 7, 8 | bitri 275 |
. . 3
⊢ (𝑋 ∈ dom 𝐹 ↔ ∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}𝑋 ∈ dom 𝑔) |
| 10 | | eqid 2737 |
. . . . . . . 8
⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
| 11 | 10 | wfrlem1OLD 8348 |
. . . . . . 7
⊢ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))} |
| 12 | 11 | eqabri 2885 |
. . . . . 6
⊢ (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))) |
| 13 | | predeq3 6325 |
. . . . . . . . . . . . 13
⊢ (𝑤 = 𝑋 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑋)) |
| 14 | 13 | sseq1d 4015 |
. . . . . . . . . . . 12
⊢ (𝑤 = 𝑋 → (Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧)) |
| 15 | 14 | rspccv 3619 |
. . . . . . . . . . 11
⊢
(∀𝑤 ∈
𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧 → (𝑋 ∈ 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧)) |
| 16 | 15 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) → (𝑋 ∈ 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧)) |
| 17 | | fndm 6671 |
. . . . . . . . . . . . 13
⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧) |
| 18 | 17 | eleq2d 2827 |
. . . . . . . . . . . 12
⊢ (𝑔 Fn 𝑧 → (𝑋 ∈ dom 𝑔 ↔ 𝑋 ∈ 𝑧)) |
| 19 | 17 | sseq2d 4016 |
. . . . . . . . . . . 12
⊢ (𝑔 Fn 𝑧 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧)) |
| 20 | 18, 19 | imbi12d 344 |
. . . . . . . . . . 11
⊢ (𝑔 Fn 𝑧 → ((𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔) ↔ (𝑋 ∈ 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧))) |
| 21 | 20 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) → ((𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔) ↔ (𝑋 ∈ 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑧))) |
| 22 | 16, 21 | mpbird 257 |
. . . . . . . . 9
⊢ ((𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔)) |
| 23 | 22 | adantrl 716 |
. . . . . . . 8
⊢ ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)) → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔)) |
| 24 | 23 | 3adant3 1133 |
. . . . . . 7
⊢ ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔)) |
| 25 | 24 | exlimiv 1930 |
. . . . . 6
⊢
(∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐺‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔)) |
| 26 | 12, 25 | sylbi 217 |
. . . . 5
⊢ (𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} → (𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔)) |
| 27 | 26 | reximia 3081 |
. . . 4
⊢
(∃𝑔 ∈
{𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}𝑋 ∈ dom 𝑔 → ∃𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔) |
| 28 | | ssiun 5046 |
. . . 4
⊢
(∃𝑔 ∈
{𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔) |
| 29 | 27, 28 | syl 17 |
. . 3
⊢
(∃𝑔 ∈
{𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}𝑋 ∈ dom 𝑔 → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔) |
| 30 | 9, 29 | sylbi 217 |
. 2
⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ 𝑔 ∈ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}dom 𝑔) |
| 31 | 30, 6 | sseqtrrdi 4025 |
1
⊢ (𝑋 ∈ dom 𝐹 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐹) |