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Theorem satfv0fun 34660
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 6582 . . 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 7418 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑖𝑔𝑗) = (𝑘𝑔𝑗))
32eqeq2d 2741 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑗)))
4 fveq2 6890 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑓𝑖) = (𝑓𝑘))
54breq1d 5157 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((𝑓𝑖)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑗)))
65rabbidv 3438 . . . . . . . . . 10 (𝑖 = 𝑘 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)})
76eqeq2d 2741 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ↔ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)}))
83, 7anbi12d 629 . . . . . . . 8 (𝑖 = 𝑘 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑘𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)})))
9 oveq2 7419 . . . . . . . . . 10 (𝑗 = 𝑙 → (𝑘𝑔𝑗) = (𝑘𝑔𝑙))
109eqeq2d 2741 . . . . . . . . 9 (𝑗 = 𝑙 → (𝑥 = (𝑘𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑙)))
11 fveq2 6890 . . . . . . . . . . . 12 (𝑗 = 𝑙 → (𝑓𝑗) = (𝑓𝑙))
1211breq2d 5159 . . . . . . . . . . 11 (𝑗 = 𝑙 → ((𝑓𝑘)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑙)))
1312rabbidv 3438 . . . . . . . . . 10 (𝑗 = 𝑙 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)})
1413eqeq2d 2741 . . . . . . . . 9 (𝑗 = 𝑙 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)} ↔ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}))
1510, 14anbi12d 629 . . . . . . . 8 (𝑗 = 𝑙 → ((𝑥 = (𝑘𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)})))
168, 15cbvrex2vw 3237 . . . . . . 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 2754 . . . . . . . . . . . . . . 15 ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑥 = (𝑘𝑔𝑙)) → (𝑖𝑔𝑗) = (𝑘𝑔𝑙))
18 goeleq12bg 34638 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑘𝑔𝑙) ↔ (𝑖 = 𝑘𝑗 = 𝑙)))
194adantr 479 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑖) = (𝑓𝑘))
2019eqcomd 2736 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑘) = (𝑓𝑖))
2111adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑗) = (𝑓𝑙))
2221eqcomd 2736 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑙) = (𝑓𝑗))
2320, 22breq12d 5160 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝑘𝑗 = 𝑙) → ((𝑓𝑘)𝐸(𝑓𝑙) ↔ (𝑓𝑖)𝐸(𝑓𝑗)))
2423rabbidv 3438 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝑘𝑗 = 𝑙) → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})
25 eqeq12 2747 . . . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝑘𝑗 = 𝑙) → ((𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) → 𝑦 = 𝑧))
2726expd 414 . . . . . . . . . . . . . . . 16 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} → (𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} → 𝑦 = 𝑧)))
2818, 27syl6bi 252 . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . 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 421 . . . . . . . . . . . 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 409 . . . . . . . . . 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 3209 . . . . . . . . 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 405 . . . . 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 1796 . . . 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 2734 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ↔ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}))
4140anbi2d 627 . . . . . 6 (𝑦 = 𝑧 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})))
42412rexbidv 3217 . . . . 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 2558 . . . 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 1799 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})}
46 eqid 2730 . . . 4 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
4746satfv0 34647 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})})
4847funeqd 6569 . 2 ((𝑀𝑉𝐸𝑊) → (Fun ((𝑀 Sat 𝐸)‘∅) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})}))
4945, 48mpbiri 257 1 ((𝑀𝑉𝐸𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wal 1537   = wceq 1539  wcel 2104  ∃*wmo 2530  wrex 3068  {crab 3430  c0 4321   class class class wbr 5147  {copab 5209  Fun wfun 6536  cfv 6542  (class class class)co 7411  ωcom 7857  m cmap 8822  𝑔cgoe 34622   Sat csat 34625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-goel 34629  df-sat 34632
This theorem is referenced by:  satffunlem1  34696  satffun  34698  satfv0fvfmla0  34702
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