Theorem satfv0fvfmla0 35913

Description: The value of the satisfaction predicate as function over a wff code at ∅. (Contributed by AV, 2-Nov-2023.)

Hypothesis

Ref	Expression
satfv0fv.s	⊢ 𝑆 = (𝑀 Sat 𝐸)

Assertion

Ref	Expression
satfv0fvfmla0	⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))})

Distinct variable groups:   𝐸,𝑎   𝑀,𝑎   𝑋,𝑎

Allowed substitution hints:   𝑆(𝑎)   𝑉(𝑎)   𝑊(𝑎)

Proof of Theorem satfv0fvfmla0

Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		satfv0fun 35871	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2		satfv0fv.s	. . . . . 6 ⊢ 𝑆 = (𝑀 Sat 𝐸)
3	2	fveq1i 6882	. . . . 5 ⊢ (𝑆‘∅) = ((𝑀 Sat 𝐸)‘∅)
4	3	funeqi 6557	. . . 4 ⊢ (Fun (𝑆‘∅) ↔ Fun ((𝑀 Sat 𝐸)‘∅))
5	1, 4	sylibr 237	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun (𝑆‘∅))
6	5	3adant3 1149	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun (𝑆‘∅))
7		fmla0 35882	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
8	7	eleq2i 2854	. . . . . . 7 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)})
9		eqeq1 2766	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑋 = (𝑖∈𝑔𝑗)))
10	9	2rexbidv 3229	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈𝑔𝑗)))
11	10	elrab 3649	. . . . . . 7 ⊢ (𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} ↔ (𝑋 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈𝑔𝑗)))
12	8, 11	bitri 278	. . . . . 6 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ (𝑋 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈𝑔𝑗)))
13		simpr 489	. . . . . . . . . 10 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈𝑔𝑗)) → 𝑋 = (𝑖∈𝑔𝑗))
14		goel 35847	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
15	14	eqeq2d 2773	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖∈𝑔𝑗) ↔ 𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
16		2fveq3 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd ‘𝑋)) = (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
17		0ex 5269	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ V
18		opex 5444	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨𝑖, 𝑗⟩ ∈ V
19	17, 18	op2nd 7993	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩) = ⟨𝑖, 𝑗⟩
20	19	fveq2i 6884	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (1st ‘⟨𝑖, 𝑗⟩)
21		vex 3458	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑖 ∈ V
22		vex 3458	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑗 ∈ V
23	21, 22	op1st 7992	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨𝑖, 𝑗⟩) = 𝑖
24	20, 23	eqtri 2785	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑖
25	16, 24	eqtrdi 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd ‘𝑋)) = 𝑖)
26	25	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(1st ‘(2nd ‘𝑋))) = (𝑎‘𝑖))
27		2fveq3 6886	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd ‘𝑋)) = (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
28	19	fveq2i 6884	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (2nd ‘⟨𝑖, 𝑗⟩)
29	21, 22	op2nd 7993	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘⟨𝑖, 𝑗⟩) = 𝑗
30	28, 29	eqtri 2785	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑗
31	27, 30	eqtrdi 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd ‘𝑋)) = 𝑗)
32	31	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(2nd ‘(2nd ‘𝑋))) = (𝑎‘𝑗))
33	26, 32	breq12d 5121	. . . . . . . . . . . . 13 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋))) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
34	15, 33	biimtrdi 256	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖∈𝑔𝑗) → ((𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋))) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗))))
35	34	imp 411	. . . . . . . . . . 11 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈𝑔𝑗)) → ((𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋))) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
36	35	rabbidv 3422	. . . . . . . . . 10 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈𝑔𝑗)) → {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})
37	13, 36	jca 520	. . . . . . . . 9 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈𝑔𝑗)) → (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
38	37	ex 417	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖∈𝑔𝑗) → (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
39	38	reximdva 3177	. . . . . . 7 ⊢ (𝑖 ∈ ω → (∃𝑗 ∈ ω 𝑋 = (𝑖∈𝑔𝑗) → ∃𝑗 ∈ ω (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
40	39	reximia 3099	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈𝑔𝑗) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
41	12, 40	simplbiim 513	. . . . 5 ⊢ (𝑋 ∈ (Fmla‘∅) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
42	41	3ad2ant3 1152	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
43		simp3 1155	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
44		ovex 7445	. . . . . 6 ⊢ (𝑀 ↑m ω) ∈ V
45	44	rabex 5308	. . . . 5 ⊢ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} ∈ V
46		eqeq1 2766	. . . . . . . 8 ⊢ (𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} → (𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} ↔ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
47	9, 46	bi2anan9 649	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}) → ((𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
48	47	2rexbidv 3229	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}) → (∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
49	48	opelopabga 5516	. . . . 5 ⊢ ((𝑋 ∈ (Fmla‘∅) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} ∈ V) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
50	43, 45, 49	sylancl 597	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})} ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖∈𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
51	42, 50	mpbird 260	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
52	2	satfv0 35858	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑆‘∅) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
53	52	eleq2d 2848	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ (𝑆‘∅) ↔ ⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}))
54	53	3adant3 1149	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ (𝑆‘∅) ↔ ⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}))
55	51, 54	mpbird 260	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ (𝑆‘∅))
56		funopfv 6930	. 2 ⊢ (Fun (𝑆‘∅) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ (𝑆‘∅) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}))
57	6, 55, 56	sylc 66	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∃wrex 3088  {crab 3415  Vcvv 3454  ∅c0 4285  ⟨cop 4594   class class class wbr 5108  {copab 5172  Fun wfun 6530  ‘cfv 6536  (class class class)co 7412  ωcom 7860  1^st c1st 7982  2^nd c2nd 7983   ↑_m cmap 8822  ∈_𝑔cgoe 35833   Sat csat 35836  Fmlacfmla 35837

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-inf2 9608

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-map 8824  df-goel 35840  df-sat 35843  df-fmla 35845

This theorem is used by:  satefvfmla0  35918