Theorem satfv0fun 32618

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 6390	. . 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 7163	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → (𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑗))
3	2	eqeq2d 2832	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑥 = (𝑘∈𝑔𝑗)))
4		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑘 → (𝑓‘𝑖) = (𝑓‘𝑘))
5	4	breq1d 5076	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑘 → ((𝑓‘𝑖)𝐸(𝑓‘𝑗) ↔ (𝑓‘𝑘)𝐸(𝑓‘𝑗)))
6	5	rabbidv 3480	. . . . . . . . . 10 ⊢ (𝑖 = 𝑘 → {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑖)𝐸(𝑓‘𝑗)} = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)})
7	6	eqeq2d 2832	. . . . . . . . 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 632	. . . . . . . 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 7164	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → (𝑘∈𝑔𝑗) = (𝑘∈𝑔𝑙))
10	9	eqeq2d 2832	. . . . . . . . 9 ⊢ (𝑗 = 𝑙 → (𝑥 = (𝑘∈𝑔𝑗) ↔ 𝑥 = (𝑘∈𝑔𝑙)))
11		fveq2 6670	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑙 → (𝑓‘𝑗) = (𝑓‘𝑙))
12	11	breq2d 5078	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑙 → ((𝑓‘𝑘)𝐸(𝑓‘𝑗) ↔ (𝑓‘𝑘)𝐸(𝑓‘𝑙)))
13	12	rabbidv 3480	. . . . . . . . . 10 ⊢ (𝑗 = 𝑙 → {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑗)} = {𝑓 ∈ (𝑀 ↑m ω) ∣ (𝑓‘𝑘)𝐸(𝑓‘𝑙)})
14	13	eqeq2d 2832	. . . . . . . . 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 632	. . . . . . . 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 3462	. . . . . . 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 2842	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑥 = (𝑘∈𝑔𝑙)) → (𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑙))
18		goeleq12bg 32596	. . . . . . . . . . . . . . . 16 ⊢ (((𝑘 ∈ ω ∧ 𝑙 ∈ ω) ∧ (𝑖 ∈ ω ∧ 𝑗 ∈ ω)) → ((𝑖∈𝑔𝑗) = (𝑘∈𝑔𝑙) ↔ (𝑖 = 𝑘 ∧ 𝑗 = 𝑙)))
19	4	adantr 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑖) = (𝑓‘𝑘))
20	19	eqcomd 2827	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑘) = (𝑓‘𝑖))
21	11	adantl 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑗) = (𝑓‘𝑙))
22	21	eqcomd 2827	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → (𝑓‘𝑙) = (𝑓‘𝑗))
23	20, 22	breq12d 5079	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 = 𝑘 ∧ 𝑗 = 𝑙) → ((𝑓‘𝑘)𝐸(𝑓‘𝑙) ↔ (𝑓‘𝑖)𝐸(𝑓‘𝑗)))
24	23	rabbidv 3480	. . . . . . . . . . . . . . . . . 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 2835	. . . . . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . 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 255	. . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 425	. . . . . . . . . . . 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 413	. . . . . . . . . 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 3294	. . . . . . . . 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 3292	. . . . . . 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 219	. . . . . 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 409	. . . . 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 2825	. . . . . . 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 630	. . . . . 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 3300	. . . . 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 2650	. . . 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 233	. . 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 2821	. . . 4 ⊢ (𝑀 Sat 𝐸) = (𝑀 Sat 𝐸)
47	46	satfv0 32605	. . 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 6377	. 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 260	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  ∃*wmo 2620  ∃wrex 3139  {crab 3142  ∅c0 4291   class class class wbr 5066  {copab 5128  Fun wfun 6349  ‘cfv 6355  (class class class)co 7156  ωcom 7580   ↑_m cmap 8406  ∈_𝑔cgoe 32580   Sat csat 32583

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-goel 32587  df-sat 32590

This theorem is referenced by:  satffunlem1  32654  satffun  32656  satfv0fvfmla0  32660