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Theorem satfv0fun 34362
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 6584 . . 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 7416 . . . . . . . . . 10 (𝑖 = 𝑘 → (𝑖𝑔𝑗) = (𝑘𝑔𝑗))
32eqeq2d 2744 . . . . . . . . 9 (𝑖 = 𝑘 → (𝑥 = (𝑖𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑗)))
4 fveq2 6892 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑓𝑖) = (𝑓𝑘))
54breq1d 5159 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((𝑓𝑖)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑗)))
65rabbidv 3441 . . . . . . . . . 10 (𝑖 = 𝑘 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)})
76eqeq2d 2744 . . . . . . . . 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 7417 . . . . . . . . . 10 (𝑗 = 𝑙 → (𝑘𝑔𝑗) = (𝑘𝑔𝑙))
109eqeq2d 2744 . . . . . . . . 9 (𝑗 = 𝑙 → (𝑥 = (𝑘𝑔𝑗) ↔ 𝑥 = (𝑘𝑔𝑙)))
11 fveq2 6892 . . . . . . . . . . . 12 (𝑗 = 𝑙 → (𝑓𝑗) = (𝑓𝑙))
1211breq2d 5161 . . . . . . . . . . 11 (𝑗 = 𝑙 → ((𝑓𝑘)𝐸(𝑓𝑗) ↔ (𝑓𝑘)𝐸(𝑓𝑙)))
1312rabbidv 3441 . . . . . . . . . 10 (𝑗 = 𝑙 → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑗)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)})
1413eqeq2d 2744 . . . . . . . . 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 3240 . . . . . . 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 2757 . . . . . . . . . . . . . . 15 ((𝑥 = (𝑖𝑔𝑗) ∧ 𝑥 = (𝑘𝑔𝑙)) → (𝑖𝑔𝑗) = (𝑘𝑔𝑙))
18 goeleq12bg 34340 . . . . . . . . . . . . . . . 16 (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖𝑔𝑗) = (𝑘𝑔𝑙) ↔ (𝑖 = 𝑘𝑗 = 𝑙)))
194adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑖) = (𝑓𝑘))
2019eqcomd 2739 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑘) = (𝑓𝑖))
2111adantl 483 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑗) = (𝑓𝑙))
2221eqcomd 2739 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝑘𝑗 = 𝑙) → (𝑓𝑙) = (𝑓𝑗))
2320, 22breq12d 5162 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝑘𝑗 = 𝑙) → ((𝑓𝑘)𝐸(𝑓𝑙) ↔ (𝑓𝑖)𝐸(𝑓𝑗)))
2423rabbidv 3441 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝑘𝑗 = 𝑙) → {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑘)𝐸(𝑓𝑙)} = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})
25 eqeq12 2750 . . . . . . . . . . . . . . . . . 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 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 3212 . . . . . . . . 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 3200 . . . . . . 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 2737 . . . . . . 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 3220 . . . . 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 2561 . . . 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 2733 . . . 4 (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
4746satfv0 34349 . . 3 ((𝑀𝑉𝐸𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀m ω) ∣ (𝑓𝑖)𝐸(𝑓𝑗)})})
4847funeqd 6571 . 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 2533  wrex 3071  {crab 3433  c0 4323   class class class wbr 5149  {copab 5211  Fun wfun 6538  cfv 6544  (class class class)co 7409  ωcom 7855  m cmap 8820  𝑔cgoe 34324   Sat csat 34327
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-goel 34331  df-sat 34334
This theorem is referenced by:  satffunlem1  34398  satffun  34400  satfv0fvfmla0  34404
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