| Step | Hyp | Ref
| Expression |
| 1 | | df-ov 7434 |
. . 3
⊢ (𝐴𝐹𝐵) = (𝐹‘〈𝐴, 𝐵〉) |
| 2 | | opelxp 5721 |
. . . 4
⊢
(〈𝐴, 𝐵〉 ∈ ( No × No ) ↔
(𝐴 ∈ No ∧ 𝐵 ∈ No
)) |
| 3 | | eqid 2737 |
. . . . . . 7
⊢
{〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} = {〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} |
| 4 | | eqid 2737 |
. . . . . . 7
⊢
{〈𝑎, 𝑏〉 ∣ (𝑎 ∈ (
No × No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} = {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} |
| 5 | 3, 4 | noxpordfr 27984 |
. . . . . 6
⊢
{〈𝑎, 𝑏〉 ∣ (𝑎 ∈ (
No × No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr ( No
× No ) |
| 6 | 3, 4 | noxpordpo 27983 |
. . . . . 6
⊢
{〈𝑎, 𝑏〉 ∣ (𝑎 ∈ (
No × No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po ( No
× No ) |
| 7 | 3, 4 | noxpordse 27985 |
. . . . . 6
⊢
{〈𝑎, 𝑏〉 ∣ (𝑎 ∈ (
No × No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se ( No
× No ) |
| 8 | 5, 6, 7 | 3pm3.2i 1340 |
. . . . 5
⊢
({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ (
No × No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr ( No
× No ) ∧ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po ( No
× No ) ∧ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se ( No
× No )) |
| 9 | | norec2.1 |
. . . . . . 7
⊢ 𝐹 = norec2 (𝐺) |
| 10 | | df-norec2 27982 |
. . . . . . 7
⊢ norec2
(𝐺) = frecs({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 𝐺) |
| 11 | 9, 10 | eqtri 2765 |
. . . . . 6
⊢ 𝐹 = frecs({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 𝐺) |
| 12 | 11 | fpr2 8329 |
. . . . 5
⊢
((({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ (
No × No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Fr ( No
× No ) ∧ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Po ( No
× No ) ∧ {〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))} Se ( No
× No )) ∧ 〈𝐴, 𝐵〉 ∈ ( No
× No )) → (𝐹‘〈𝐴, 𝐵〉) = (〈𝐴, 𝐵〉𝐺(𝐹 ↾ Pred({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 〈𝐴, 𝐵〉)))) |
| 13 | 8, 12 | mpan 690 |
. . . 4
⊢
(〈𝐴, 𝐵〉 ∈ ( No × No ) →
(𝐹‘〈𝐴, 𝐵〉) = (〈𝐴, 𝐵〉𝐺(𝐹 ↾ Pred({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 〈𝐴, 𝐵〉)))) |
| 14 | 2, 13 | sylbir 235 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (𝐹‘〈𝐴, 𝐵〉) = (〈𝐴, 𝐵〉𝐺(𝐹 ↾ Pred({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 〈𝐴, 𝐵〉)))) |
| 15 | 1, 14 | eqtrid 2789 |
. 2
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (𝐴𝐹𝐵) = (〈𝐴, 𝐵〉𝐺(𝐹 ↾ Pred({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 〈𝐴, 𝐵〉)))) |
| 16 | 3, 4 | noxpordpred 27986 |
. . . 4
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → Pred({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 〈𝐴, 𝐵〉) = ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})) |
| 17 | 16 | reseq2d 5997 |
. . 3
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (𝐹 ↾ Pred({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 〈𝐴, 𝐵〉)) = (𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉}))) |
| 18 | 17 | oveq2d 7447 |
. 2
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (〈𝐴, 𝐵〉𝐺(𝐹 ↾ Pred({〈𝑎, 𝑏〉 ∣ (𝑎 ∈ ( No
× No ) ∧ 𝑏 ∈ ( No
× No ) ∧ (((1st ‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (1st ‘𝑏) ∨ (1st ‘𝑎) = (1st ‘𝑏)) ∧ ((2nd
‘𝑎){〈𝑐, 𝑑〉 ∣ 𝑐 ∈ (( L ‘𝑑) ∪ ( R ‘𝑑))} (2nd ‘𝑏) ∨ (2nd ‘𝑎) = (2nd ‘𝑏)) ∧ 𝑎 ≠ 𝑏))}, ( No
× No ), 〈𝐴, 𝐵〉))) = (〈𝐴, 𝐵〉𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})))) |
| 19 | 15, 18 | eqtrd 2777 |
1
⊢ ((𝐴 ∈
No ∧ 𝐵 ∈
No ) → (𝐴𝐹𝐵) = (〈𝐴, 𝐵〉𝐺(𝐹 ↾ ((((( L ‘𝐴) ∪ ( R ‘𝐴)) ∪ {𝐴}) × ((( L ‘𝐵) ∪ ( R ‘𝐵)) ∪ {𝐵})) ∖ {〈𝐴, 𝐵〉})))) |