| Step | Hyp | Ref
| Expression |
| 1 | | df-sat 35348 |
. . 3
⊢ Sat =
(𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω)) |
| 2 | 1 | a1i 11 |
. 2
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Sat = (𝑚 ∈ V, 𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω))) |
| 3 | | oveq1 7438 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = 𝑀 → (𝑚 ↑m ω) = (𝑀 ↑m
ω)) |
| 4 | 3 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑚 ↑m ω) = (𝑀 ↑m
ω)) |
| 5 | 4 | difeq1d 4125 |
. . . . . . . . . . . . 13
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣))) = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) |
| 6 | 5 | eqeq2d 2748 |
. . . . . . . . . . . 12
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣))) ↔ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣))))) |
| 7 | 6 | anbi2d 630 |
. . . . . . . . . . 11
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ↔ (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))) |
| 8 | 7 | rexbidv 3179 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ↔ ∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))))) |
| 9 | | simpl 482 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → 𝑚 = 𝑀) |
| 10 | 9 | raleqdv 3326 |
. . . . . . . . . . . . . 14
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢) ↔ ∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢))) |
| 11 | 4, 10 | rabeqbidv 3455 |
. . . . . . . . . . . . 13
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}) |
| 12 | 11 | eqeq2d 2748 |
. . . . . . . . . . . 12
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)} ↔ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) |
| 13 | 12 | anbi2d 630 |
. . . . . . . . . . 11
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}) ↔ (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))) |
| 14 | 13 | rexbidv 3179 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}) ↔ ∃𝑖 ∈ ω (𝑥 =
∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))) |
| 15 | 8, 14 | orbi12d 919 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ↔ (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})))) |
| 16 | 15 | rexbidv 3179 |
. . . . . . . 8
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})) ↔ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)})))) |
| 17 | 16 | opabbidv 5209 |
. . . . . . 7
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))} = {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) |
| 18 | 17 | uneq2d 4168 |
. . . . . 6
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}) = (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})) |
| 19 | 18 | mpteq2dv 5244 |
. . . . 5
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})) = (𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))}))) |
| 20 | | breq 5145 |
. . . . . . . . . . 11
⊢ (𝑒 = 𝐸 → ((𝑎‘𝑖)𝑒(𝑎‘𝑗) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗))) |
| 21 | 20 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑎‘𝑖)𝑒(𝑎‘𝑗) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗))) |
| 22 | 4, 21 | rabeqbidv 3455 |
. . . . . . . . 9
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) |
| 23 | 22 | eqeq2d 2748 |
. . . . . . . 8
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)} ↔ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})) |
| 24 | 23 | anbi2d 630 |
. . . . . . 7
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)}) ↔ (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))) |
| 25 | 24 | 2rexbidv 3222 |
. . . . . 6
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))) |
| 26 | 25 | opabbidv 5209 |
. . . . 5
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})} = {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) |
| 27 | | rdgeq12 8453 |
. . . . 5
⊢ (((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})) = (𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})) ∧ {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})} = {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) → rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) = rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})) |
| 28 | 19, 26, 27 | syl2anc 584 |
. . . 4
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) = rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})) |
| 29 | 28 | reseq1d 5996 |
. . 3
⊢ ((𝑚 = 𝑀 ∧ 𝑒 = 𝐸) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)) |
| 30 | 29 | adantl 481 |
. 2
⊢ (((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) ∧ (𝑚 = 𝑀 ∧ 𝑒 = 𝐸)) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑚 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣
∀𝑧 ∈ 𝑚 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑚 ↑m ω) ∣ (𝑎‘𝑖)𝑒(𝑎‘𝑗)})}) ↾ suc ω) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)) |
| 31 | | elex 3501 |
. . 3
⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) |
| 32 | 31 | adantr 480 |
. 2
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝑀 ∈ V) |
| 33 | | elex 3501 |
. . 3
⊢ (𝐸 ∈ 𝑊 → 𝐸 ∈ V) |
| 34 | 33 | adantl 481 |
. 2
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 𝐸 ∈ V) |
| 35 | | rdgfun 8456 |
. . 3
⊢ Fun
rec((𝑓 ∈ V ↦
(𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) |
| 36 | | omex 9683 |
. . . . 5
⊢ ω
∈ V |
| 37 | 36 | sucex 7826 |
. . . 4
⊢ suc
ω ∈ V |
| 38 | 37 | a1i 11 |
. . 3
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → suc ω ∈
V) |
| 39 | | resfunexg 7235 |
. . 3
⊢ ((Fun
rec((𝑓 ∈ V ↦
(𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ∧ suc ω ∈ V) →
(rec((𝑓 ∈ V ↦
(𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω) ∈
V) |
| 40 | 35, 38, 39 | sylancr 587 |
. 2
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω) ∈
V) |
| 41 | 2, 30, 32, 34, 40 | ovmpod 7585 |
1
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st
‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖
((2nd ‘𝑢)
∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣
∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)) |