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Theorem satfv0fun 33965
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 6536 . . 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 7364 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑖𝑔𝑗) = (𝑘𝑔𝑗))
32eqeq2d 2747 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑗)))
4 fveq2 6842 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑓𝑖) = (𝑓𝑘))
54breq1d 5115 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((𝑓𝑖)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑗)))
65rabbidv 3415 . . . . . . . . . 10 (𝑖 = 𝑘 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)})
76eqeq2d 2747 . . . . . . . . 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 7365 . . . . . . . . . 10 (𝑗 = 𝑙 → (𝑘𝑔𝑗) = (𝑘𝑔𝑙))
109eqeq2d 2747 . . . . . . . . 9 (𝑗 = 𝑙 → (𝑥 = (𝑘𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑙)))
11 fveq2 6842 . . . . . . . . . . . 12 (𝑗 = 𝑙 → (𝑓𝑗) = (𝑓𝑙))
1211breq2d 5117 . . . . . . . . . . 11 (𝑗 = 𝑙 → ((𝑓𝑘)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑙)))
1312rabbidv 3415 . . . . . . . . . 10 (𝑗 = 𝑙 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)})
1413eqeq2d 2747 . . . . . . . . 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 3228 . . . . . . 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 2760 . . . . . . . . . . . . . . 15 ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑥 = (𝑘𝑔𝑙)) → (𝑖𝑔𝑗) = (𝑘𝑔𝑙))
18 goeleq12bg 33943 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑘𝑔𝑙) ↔ (𝑖 = 𝑘𝑗 = 𝑙)))
194adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑖) = (𝑓𝑘))
2019eqcomd 2742 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑘) = (𝑓𝑖))
2111adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑗) = (𝑓𝑙))
2221eqcomd 2742 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑙) = (𝑓𝑗))
2320, 22breq12d 5118 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝑘𝑗 = 𝑙) → ((𝑓𝑘)𝐸(𝑓𝑙) ↔ (𝑓𝑖)𝐸(𝑓𝑗)))
2423rabbidv 3415 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝑘𝑗 = 𝑙) → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})
25 eqeq12 2753 . . . . . . . . . . . . . . . . . 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 3205 . . . . . . . . 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 3196 . . . . . . 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 2740 . . . . . . 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 3213 . . . . 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 2564 . . . 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 2736 . . . 4 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
4746satfv0 33952 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})})
4847funeqd 6523 . 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 2536  wrex 3073  {crab 3407  c0 4282   class class class wbr 5105  {copab 5167  Fun wfun 6490  cfv 6496  (class class class)co 7357  ωcom 7802  m cmap 8765  𝑔cgoe 33927   Sat csat 33930
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-goel 33934  df-sat 33937
This theorem is referenced by:  satffunlem1  34001  satffun  34003  satfv0fvfmla0  34007
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