Theorem satfv0fun 33233

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 6453	. . 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 7262	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑗))
3	2	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = (𝑘∈𝑔𝑗)))
4		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (𝑓‘𝑖) = (𝑓‘𝑘))
5	4	breq1d 5080	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → ((𝑓‘𝑖)𝐸(𝑓‘𝑗) ↔ (𝑓‘𝑘)𝐸(𝑓‘𝑗)))
6	5	rabbidv 3404	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)})
7	6	eqeq2d 2749	. . . . . . . . 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 7263	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → (𝑘∈𝑔𝑗) = (𝑘∈𝑔𝑙))
10	9	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝑥 = (𝑘∈𝑔𝑗) ↔ 𝑥 = (𝑘∈𝑔𝑙)))
11		fveq2 6756	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑙 → (𝑓‘𝑗) = (𝑓‘𝑙))
12	11	breq2d 5082	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑙 → ((𝑓‘𝑘)𝐸(𝑓‘𝑗) ↔ (𝑓‘𝑘)𝐸(𝑓‘𝑙)))
13	12	rabbidv 3404	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)} = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)})
14	13	eqeq2d 2749	. . . . . . . . 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 3386	. . . . . . 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 2762	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑥 = (𝑘∈𝑔𝑙)) → (𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑙))
18		goeleq12bg 33211	. . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑙) = (𝑓‘𝑗))
23	20, 22	breq12d 5083	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → ((𝑓‘𝑘)𝐸(𝑓‘𝑙) ↔ (𝑓‘𝑖)𝐸(𝑓‘𝑗)))
24	23	rabbidv 3404	. . . . . . . . . . . . . . . . . 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 2755	. . . . . . . . . . . . . . . . . 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 252	. . . . . . . . . . . . . . 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 3222	. . . . . . . . 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 3220	. . . . . . 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 1800	. . . 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 2742	. . . . . . 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 3228	. . . . 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 2566	. . . 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 1803	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)})}
46		eqid 2738	. . . 4 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
47	46	satfv0 33220	. . 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 6440	. 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 257	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∃*wmo 2538  ∃wrex 3064  {crab 3067  ∅c0 4253   class class class wbr 5070  {copab 5132  Fun wfun 6412  ‘cfv 6418  (class class class)co 7255  ωcom 7687   ↑_m cmap 8573  ∈_𝑔cgoe 33195   Sat csat 33198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-goel 33202  df-sat 33205

This theorem is referenced by:  satffunlem1  33269  satffun  33271  satfv0fvfmla0  33275