Theorem satfv0fvfmla0 35588

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 35546	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2		satfv0fv.s	. . . . . 6 ⊢ 𝑆 = (𝑀 Sat 𝐸)
3	2	fveq1i 6836	. . . . 5 ⊢ (𝑆‘∅) = ((𝑀 Sat 𝐸)‘∅)
4	3	funeqi 6514	. . . 4 ⊢ (Fun (𝑆‘∅) ↔ Fun ((𝑀 Sat 𝐸)‘∅))
5	1, 4	sylibr 234	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun (𝑆‘∅))
6	5	3adant3 1133	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun (𝑆‘∅))
7		fmla0 35557	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)}
8	7	eleq2i 2829	. . . . . . 7 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)})
9		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 = (𝑖∈𝑔𝑗) ↔ 𝑋 = (𝑖∈𝑔𝑗)))
10	9	2rexbidv 3202	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈𝑔𝑗)))
11	10	elrab 3647	. . . . . . 7 ⊢ (𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗)} ↔ (𝑋 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈𝑔𝑗)))
12	8, 11	bitri 275	. . . . . 6 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ (𝑋 ∈ V ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈𝑔𝑗)))
13		simpr 484	. . . . . . . . . 10 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈𝑔𝑗)) → 𝑋 = (𝑖∈𝑔𝑗))
14		goel 35522	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
15	14	eqeq2d 2748	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖∈𝑔𝑗) ↔ 𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
16		2fveq3 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd ‘𝑋)) = (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
17		0ex 5253	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ V
18		opex 5413	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨𝑖, 𝑗⟩ ∈ V
19	17, 18	op2nd 7944	. . . . . . . . . . . . . . . . . 18 ⊢ (2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩) = ⟨𝑖, 𝑗⟩
20	19	fveq2i 6838	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (1st ‘⟨𝑖, 𝑗⟩)
21		vex 3445	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑖 ∈ V
22		vex 3445	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑗 ∈ V
23	21, 22	op1st 7943	. . . . . . . . . . . . . . . . 17 ⊢ (1st ‘⟨𝑖, 𝑗⟩) = 𝑖
24	20, 23	eqtri 2760	. . . . . . . . . . . . . . . 16 ⊢ (1st ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑖
25	16, 24	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1st ‘(2nd ‘𝑋)) = 𝑖)
26	25	fveq2d 6839	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(1st ‘(2nd ‘𝑋))) = (𝑎‘𝑖))
27		2fveq3 6840	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd ‘𝑋)) = (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
28	19	fveq2i 6838	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (2nd ‘⟨𝑖, 𝑗⟩)
29	21, 22	op2nd 7944	. . . . . . . . . . . . . . . . 17 ⊢ (2nd ‘⟨𝑖, 𝑗⟩) = 𝑗
30	28, 29	eqtri 2760	. . . . . . . . . . . . . . . 16 ⊢ (2nd ‘(2nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑗
31	27, 30	eqtrdi 2788	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2nd ‘(2nd ‘𝑋)) = 𝑗)
32	31	fveq2d 6839	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(2nd ‘(2nd ‘𝑋))) = (𝑎‘𝑗))
33	26, 32	breq12d 5112	. . . . . . . . . . . . 13 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋))) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
34	15, 33	biimtrdi 253	. . . . . . . . . . . 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 406	. . . . . . . . . . 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 3407	. . . . . . . . . 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 511	. . . . . . . . 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 412	. . . . . . . 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 3150	. . . . . . 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 3072	. . . . . 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 504	. . . . 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 1136	. . . 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 1139	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
44		ovex 7393	. . . . . 6 ⊢ (𝑀 ↑m ω) ∈ V
45	44	rabex 5285	. . . . 5 ⊢ {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))} ∈ V
46		eqeq1 2741	. . . . . . . 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 639	. . . . . . 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 3202	. . . . . 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 5482	. . . . 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 587	. . . 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 257	. . 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 35533	. . . . 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 2823	. . . 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 1133	. . 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 257	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ⟨𝑋, {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))}⟩ ∈ (𝑆‘∅))
56		funopfv 6884	. 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 65	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘(1st ‘(2nd ‘𝑋)))𝐸(𝑎‘(2nd ‘(2nd ‘𝑋)))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {crab 3400  Vcvv 3441  ∅c0 4286  ⟨cop 4587   class class class wbr 5099  {copab 5161  Fun wfun 6487  ‘cfv 6493  (class class class)co 7360  ωcom 7810  1^st c1st 7933  2^nd c2nd 7934   ↑_m cmap 8767  ∈_𝑔cgoe 35508   Sat csat 35511  Fmlacfmla 35512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682  ax-inf2 9554

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-map 8769  df-goel 35515  df-sat 35518  df-fmla 35520

This theorem is referenced by:  satefvfmla0  35593