Theorem satfv0fun 34828

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	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → 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.

Step	Hyp	Ref	Expression
1		funopab 6583	. . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})} ↔ ∀𝑥∃*𝑦∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}))
2		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑗))
3	2	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = (𝑘∈𝑔𝑗)))
4		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (𝑓‘𝑖) = (𝑓‘𝑘))
5	4	breq1d 5158	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → ((𝑓‘𝑖)𝐸(𝑓‘𝑗) ↔ (𝑓‘𝑘)𝐸(𝑓‘𝑗)))
6	5	rabbidv 3439	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)})
7	6	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} ↔ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)}))
8	3, 7	anbi12d 630	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) ↔ (𝑥 = (𝑘∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)})))
9		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → (𝑘∈𝑔𝑗) = (𝑘∈𝑔𝑙))
10	9	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝑥 = (𝑘∈𝑔𝑗) ↔ 𝑥 = (𝑘∈𝑔𝑙)))
11		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑙 → (𝑓‘𝑗) = (𝑓‘𝑙))
12	11	breq2d 5160	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑙 → ((𝑓‘𝑘)𝐸(𝑓‘𝑗) ↔ (𝑓‘𝑘)𝐸(𝑓‘𝑙)))
13	12	rabbidv 3439	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)} = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)})
14	13	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)} ↔ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)}))
15	10, 14	anbi12d 630	. . . . . . . 8 ⊢ (𝑗 = 𝑙 → ((𝑥 = (𝑘∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)}) ↔ (𝑥 = (𝑘∈𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)})))
16	8, 15	cbvrex2vw 3238	. . . . . . 7 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) ↔ ∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = (𝑘∈𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)}))
17		eqtr2 2755	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑥 = (𝑘∈𝑔𝑙)) → (𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑙))
18		goeleq12bg 34806	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑙) ↔ (𝑖 = 𝑘 ∧ 𝑗 = 𝑙)))
19	4	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑖) = (𝑓‘𝑘))
20	19	eqcomd 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑘) = (𝑓‘𝑖))
21	11	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑗) = (𝑓‘𝑙))
22	21	eqcomd 2737	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑙) = (𝑓‘𝑗))
23	20, 22	breq12d 5161	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → ((𝑓‘𝑘)𝐸(𝑓‘𝑙) ↔ (𝑓‘𝑖)𝐸(𝑓‘𝑗)))
24	23	rabbidv 3439	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)} = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})
25		eqeq12 2748	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)} ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) → (𝑦 = 𝑧 ↔ {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)} = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}))
26	24, 25	syl5ibrcom 246	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → ((𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)} ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) → 𝑦 = 𝑧))
27	26	expd 415	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)} → (𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} → 𝑦 = 𝑧)))
28	18, 27	syl6bi 253	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑙) → (𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)} → (𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} → 𝑦 = 𝑧))))
29	17, 28	syl5 34	. . . . . . . . . . . . . 14 ⊢ (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑥 = (𝑘∈𝑔𝑙)) → (𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)} → (𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} → 𝑦 = 𝑧))))
30	29	expd 415	. . . . . . . . . . . . 13 ⊢ (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑥 = (𝑖∈𝑔𝑗) → (𝑥 = (𝑘∈𝑔𝑙) → (𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)} → (𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} → 𝑦 = 𝑧)))))
31	30	imp4a 422	. . . . . . . . . . . 12 ⊢ (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑥 = (𝑖∈𝑔𝑗) → ((𝑥 = (𝑘∈𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)}) → (𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} → 𝑦 = 𝑧))))
32	31	com34 91	. . . . . . . . . . 11 ⊢ (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → (𝑥 = (𝑖∈𝑔𝑗) → (𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} → ((𝑥 = (𝑘∈𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)}) → 𝑦 = 𝑧))))
33	32	impd 410	. . . . . . . . . 10 ⊢ (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) → ((𝑥 = (𝑘∈𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)}) → 𝑦 = 𝑧)))
34	33	rexlimdvva 3210	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑙 ∈ ω) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) → ((𝑥 = (𝑘∈𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)}) → 𝑦 = 𝑧)))
35	34	com23 86	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ 𝑙 ∈ ω) → ((𝑥 = (𝑘∈𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) → 𝑦 = 𝑧)))
36	35	rexlimivv 3198	. . . . . . 7 ⊢ (∃𝑘 ∈ ω ∃𝑙 ∈ ω (𝑥 = (𝑘∈𝑔𝑙) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) → 𝑦 = 𝑧))
37	16, 36	sylbi 216	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) → 𝑦 = 𝑧))
38	37	imp 406	. . . . 5 ⊢ ((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})) → 𝑦 = 𝑧)
39	38	gen2 1797	. . . 4 ⊢ ∀𝑦∀𝑧((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})) → 𝑦 = 𝑧)
40		eqeq1 2735	. . . . . . 7 ⊢ (𝑦 = 𝑧 → (𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} ↔ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}))
41	40	anbi2d 628	. . . . . 6 ⊢ (𝑦 = 𝑧 → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) ↔ (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})))
42	41	2rexbidv 3218	. . . . 5 ⊢ (𝑦 = 𝑧 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})))
43	42	mo4 2559	. . . 4 ⊢ (∃*𝑦∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) ↔ ∀𝑦∀𝑧((∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)}) ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑧 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})) → 𝑦 = 𝑧))
44	39, 43	mpbir 230	. . 3 ⊢ ∃*𝑦∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})
45	1, 44	mpgbir 1800	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})}
46		eqid 2731	. . . 4 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
47	46	satfv0 34815	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸)‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})})
48	47	funeqd 6570	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (Fun ((𝑀 Sat 𝐸)‘∅) ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})}))
49	45, 48	mpbiri 258	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2105  ∃*wmo 2531  ∃wrex 3069  {crab 3431  ∅c0 4322   class class class wbr 5148  {copab 5210  Fun wfun 6537  ‘cfv 6543  (class class class)co 7412  ωcom 7859   ↑_m cmap 8826  ∈_𝑔cgoe 34790   Sat csat 34793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-inf2 9642

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-goel 34797  df-sat 34800

This theorem is referenced by:  satffunlem1  34864  satffun  34866  satfv0fvfmla0  34870