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Theorem satfv0fun 34177
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 6569 . . 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 7397 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑖𝑔𝑗) = (𝑘𝑔𝑗))
32eqeq2d 2742 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑗)))
4 fveq2 6875 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑓𝑖) = (𝑓𝑘))
54breq1d 5148 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((𝑓𝑖)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑗)))
65rabbidv 3437 . . . . . . . . . 10 (𝑖 = 𝑘 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)})
76eqeq2d 2742 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ↔ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)}))
83, 7anbi12d 631 . . . . . . . 8 (𝑖 = 𝑘 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑘𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)})))
9 oveq2 7398 . . . . . . . . . 10 (𝑗 = 𝑙 → (𝑘𝑔𝑗) = (𝑘𝑔𝑙))
109eqeq2d 2742 . . . . . . . . 9 (𝑗 = 𝑙 → (𝑥 = (𝑘𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑙)))
11 fveq2 6875 . . . . . . . . . . . 12 (𝑗 = 𝑙 → (𝑓𝑗) = (𝑓𝑙))
1211breq2d 5150 . . . . . . . . . . 11 (𝑗 = 𝑙 → ((𝑓𝑘)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑙)))
1312rabbidv 3437 . . . . . . . . . 10 (𝑗 = 𝑙 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)})
1413eqeq2d 2742 . . . . . . . . 9 (𝑗 = 𝑙 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)} ↔ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)}))
1510, 14anbi12d 631 . . . . . . . 8 (𝑗 = 𝑙 → ((𝑥 = (𝑘𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑘𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)})))
168, 15cbvrex2vw 3238 . . . . . . 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 2755 . . . . . . . . . . . . . . 15 ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑥 = (𝑘𝑔𝑙)) → (𝑖𝑔𝑗) = (𝑘𝑔𝑙))
18 goeleq12bg 34155 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑘𝑔𝑙) ↔ (𝑖 = 𝑘𝑗 = 𝑙)))
194adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑖) = (𝑓𝑘))
2019eqcomd 2737 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑘) = (𝑓𝑖))
2111adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑗) = (𝑓𝑙))
2221eqcomd 2737 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑙) = (𝑓𝑗))
2320, 22breq12d 5151 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝑘𝑗 = 𝑙) → ((𝑓𝑘)𝐸(𝑓𝑙) ↔ (𝑓𝑖)𝐸(𝑓𝑗)))
2423rabbidv 3437 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝑘𝑗 = 𝑙) → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})
25 eqeq12 2748 . . . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . 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 423 . . . . . . . . . . . 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 411 . . . . . . . . . 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 3210 . . . . . . . . 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 3198 . . . . . . 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 407 . . . . 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 1798 . . . 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 2735 . . . . . . 7 (𝑦 = 𝑧 → (𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} ↔ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}))
4140anbi2d 629 . . . . . 6 (𝑦 = 𝑧 → ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)}) ↔ (𝑥 = (𝑖𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})))
42412rexbidv 3218 . . . . 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 2559 . . . 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 1801 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})}
46 eqid 2731 . . . 4 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
4746satfv0 34164 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})})
4847funeqd 6556 . 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 396  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2531  wrex 3069  {crab 3429  c0 4315   class class class wbr 5138  {copab 5200  Fun wfun 6523  cfv 6529  (class class class)co 7390  ωcom 7835  m cmap 8800  𝑔cgoe 34139   Sat csat 34142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7705  ax-inf2 9615
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3430  df-v 3472  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4520  df-pw 4595  df-sn 4620  df-pr 4622  df-op 4626  df-uni 4899  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6286  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6531  df-fn 6532  df-f 6533  df-f1 6534  df-fo 6535  df-f1o 6536  df-fv 6537  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7836  df-2nd 7955  df-frecs 8245  df-wrecs 8276  df-recs 8350  df-rdg 8389  df-goel 34146  df-sat 34149
This theorem is referenced by:  satffunlem1  34213  satffun  34215  satfv0fvfmla0  34219
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