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Theorem satfv0fun 34005
Description: The value of the satisfaction predicate as function over wff codes at is a function. (Contributed by AV, 15-Oct-2023.)
Assertion
Ref Expression
satfv0fun ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))

Proof of Theorem satfv0fun
Dummy variables 𝑓 𝑖 𝑗 𝑘 𝑙 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funopab 6541 . . 3 (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})} ↔ ∀𝑥∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}))
2 oveq1 7369 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑖𝑔𝑗) = (𝑘𝑔𝑗))
32eqeq2d 2748 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑗)))
4 fveq2 6847 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑓𝑖) = (𝑓𝑘))
54breq1d 5120 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((𝑓𝑖)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑗)))
65rabbidv 3418 . . . . . . . . . 10 (𝑖 = 𝑘 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)})
76eqeq2d 2748 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ↔ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)}))
83, 7anbi12d 632 . . . . . . . 8 (𝑖 = 𝑘 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑘𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)})))
9 oveq2 7370 . . . . . . . . . 10 (𝑗 = 𝑙 → (𝑘𝑔𝑗) = (𝑘𝑔𝑙))
109eqeq2d 2748 . . . . . . . . 9 (𝑗 = 𝑙 → (𝑥 = (𝑘𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑙)))
11 fveq2 6847 . . . . . . . . . . . 12 (𝑗 = 𝑙 → (𝑓𝑗) = (𝑓𝑙))
1211breq2d 5122 . . . . . . . . . . 11 (𝑗 = 𝑙 → ((𝑓𝑘)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑙)))
1312rabbidv 3418 . . . . . . . . . 10 (𝑗 = 𝑙 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)})
1413eqeq2d 2748 . . . . . . . . 9 (𝑗 = 𝑙 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)} ↔ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}))
1510, 14anbi12d 632 . . . . . . . 8 (𝑗 = 𝑙 → ((𝑥 = (𝑘𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)})))
168, 15cbvrex2vw 3231 . . . . . . 7 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}))
17 eqtr2 2761 . . . . . . . . . . . . . . 15 ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑥 = (𝑘𝑔𝑙)) → (𝑖𝑔𝑗) = (𝑘𝑔𝑙))
18 goeleq12bg 33983 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑘𝑔𝑙) ↔ (𝑖 = 𝑘𝑗 = 𝑙)))
194adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑖) = (𝑓𝑘))
2019eqcomd 2743 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑘) = (𝑓𝑖))
2111adantl 483 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑗) = (𝑓𝑙))
2221eqcomd 2743 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑙) = (𝑓𝑗))
2320, 22breq12d 5123 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝑘𝑗 = 𝑙) → ((𝑓𝑘)𝐸(𝑓𝑙) ↔ (𝑓𝑖)𝐸(𝑓𝑗)))
2423rabbidv 3418 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝑘𝑗 = 𝑙) → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})
25 eqeq12 2754 . . . . . . . . . . . . . . . . . 18 ((𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → (𝑦 = 𝑧 ↔ {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}))
2624, 25syl5ibrcom 247 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝑘𝑗 = 𝑙) → ((𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑦 = 𝑧))
2726expd 417 . . . . . . . . . . . . . . . 16 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → 𝑦 = 𝑧)))
2818, 27syl6bi 253 . . . . . . . . . . . . . . 15 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑘𝑔𝑙) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → 𝑦 = 𝑧))))
2917, 28syl5 34 . . . . . . . . . . . . . 14 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑥 = (𝑘𝑔𝑙)) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → 𝑦 = 𝑧))))
3029expd 417 . . . . . . . . . . . . 13 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑥 = (𝑖𝑔𝑗) → (𝑥 = (𝑘𝑔𝑙) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → 𝑦 = 𝑧)))))
3130imp4a 424 . . . . . . . . . . . 12 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑥 = (𝑖𝑔𝑗) → ((𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}) → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → 𝑦 = 𝑧))))
3231com34 91 . . . . . . . . . . 11 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑥 = (𝑖𝑔𝑗) → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → ((𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}) → 𝑦 = 𝑧))))
3332impd 412 . . . . . . . . . 10 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → ((𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}) → 𝑦 = 𝑧)))
3433rexlimdvva 3206 . . . . . . . . 9 ((𝑘 ∈ ω ∧ 𝑙 ∈ ω) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → ((𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}) → 𝑦 = 𝑧)))
3534com23 86 . . . . . . . 8 ((𝑘 ∈ ω ∧ 𝑙 ∈ ω) → ((𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑦 = 𝑧)))
3635rexlimivv 3197 . . . . . . 7 (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑦 = 𝑧))
3716, 36sylbi 216 . . . . . 6 (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑦 = 𝑧))
3837imp 408 . . . . 5 ((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})) → 𝑦 = 𝑧)
3938gen2 1799 . . . 4 𝑦𝑧((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})) → 𝑦 = 𝑧)
40 eqeq1 2741 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ↔ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}))
4140anbi2d 630 . . . . . 6 (𝑦 = 𝑧 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})))
42412rexbidv 3214 . . . . 5 (𝑦 = 𝑧 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})))
4342mo4 2565 . . . 4 (∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ ∀𝑦𝑧((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})) → 𝑦 = 𝑧))
4439, 43mpbir 230 . . 3 ∃*𝑦𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})
451, 44mpgbir 1802 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})}
46 eqid 2737 . . . 4 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
4746satfv0 33992 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})})
4847funeqd 6528 . 2 ((𝑀𝑉𝐸𝑊) → (Fun ((𝑀 Sat 𝐸)‘∅) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})}))
4945, 48mpbiri 258 1 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540   = wceq 1542  wcel 2107  ∃*wmo 2537  wrex 3074  {crab 3410  c0 4287   class class class wbr 5110  {copab 5172  Fun wfun 6495  cfv 6501  (class class class)co 7362  ωcom 7807  m cmap 8772  𝑔cgoe 33967   Sat csat 33970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-inf2 9584
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-goel 33974  df-sat 33977
This theorem is referenced by:  satffunlem1  34041  satffun  34043  satfv0fvfmla0  34047
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