satfv0fvfmla0 - Mathbox for Mario Carneiro

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Theorem satfv0fvfmla0 34404

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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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 34362	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun ((𝑀 Sat 𝐸)‘∅))
2		satfv0fv.s	. . . . . 6 ⊢ 𝑆 = (𝑀 Sat 𝐸)
3	2	fveq1i 6893	. . . . 5 ⊢ (𝑆‘∅) = ((𝑀 Sat 𝐸)‘∅)
4	3	funeqi 6570	. . . 4 ⊢ (Fun (𝑆‘∅) ↔ Fun ((𝑀 Sat 𝐸)‘∅))
5	1, 4	sylibr 233	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → Fun (𝑆‘∅))
6	5	3adant3 1133	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → Fun (𝑆‘∅))
7		fmla0 34373	. . . . . . . 8 ⊢ (Fmla‘∅) = {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)}
8	7	eleq2i 2826	. . . . . . 7 ⊢ (𝑋 ∈ (Fmla‘∅) ↔ 𝑋 ∈ {𝑥 ∈ V ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗)})
9		eqeq1 2737	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 = (𝑖∈_𝑔𝑗) ↔ 𝑋 = (𝑖∈_𝑔𝑗)))
10	9	2rexbidv 3220	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈_𝑔𝑗) ↔ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈_𝑔𝑗)))
11	10	elrab 3684	. . . . . . 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 486	. . . . . . . . . 10 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈_𝑔𝑗)) → 𝑋 = (𝑖∈_𝑔𝑗))
14		goel 34338	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑖∈_𝑔𝑗) = ⟨∅, ⟨𝑖, 𝑗⟩⟩)
15	14	eqeq2d 2744	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖∈_𝑔𝑗) ↔ 𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩))
16		2fveq3 6897	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1^st ‘(2^nd ‘𝑋)) = (1^st ‘(2^nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
17		0ex 5308	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ V
18		opex 5465	. . . . . . . . . . . . . . . . . . 19 ⊢ ⟨𝑖, 𝑗⟩ ∈ V
19	17, 18	op2nd 7984	. . . . . . . . . . . . . . . . . 18 ⊢ (2^nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩) = ⟨𝑖, 𝑗⟩
20	19	fveq2i 6895	. . . . . . . . . . . . . . . . 17 ⊢ (1^st ‘(2^nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (1^st ‘⟨𝑖, 𝑗⟩)
21		vex 3479	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑖 ∈ V
22		vex 3479	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑗 ∈ V
23	21, 22	op1st 7983	. . . . . . . . . . . . . . . . 17 ⊢ (1^st ‘⟨𝑖, 𝑗⟩) = 𝑖
24	20, 23	eqtri 2761	. . . . . . . . . . . . . . . 16 ⊢ (1^st ‘(2^nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑖
25	16, 24	eqtrdi 2789	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (1^st ‘(2^nd ‘𝑋)) = 𝑖)
26	25	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(1^st ‘(2^nd ‘𝑋))) = (𝑎‘𝑖))
27		2fveq3 6897	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2^nd ‘(2^nd ‘𝑋)) = (2^nd ‘(2^nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)))
28	19	fveq2i 6895	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘(2^nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = (2^nd ‘⟨𝑖, 𝑗⟩)
29	21, 22	op2nd 7984	. . . . . . . . . . . . . . . . 17 ⊢ (2^nd ‘⟨𝑖, 𝑗⟩) = 𝑗
30	28, 29	eqtri 2761	. . . . . . . . . . . . . . . 16 ⊢ (2^nd ‘(2^nd ‘⟨∅, ⟨𝑖, 𝑗⟩⟩)) = 𝑗
31	27, 30	eqtrdi 2789	. . . . . . . . . . . . . . 15 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (2^nd ‘(2^nd ‘𝑋)) = 𝑗)
32	31	fveq2d 6896	. . . . . . . . . . . . . 14 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → (𝑎‘(2^nd ‘(2^nd ‘𝑋))) = (𝑎‘𝑗))
33	26, 32	breq12d 5162	. . . . . . . . . . . . 13 ⊢ (𝑋 = ⟨∅, ⟨𝑖, 𝑗⟩⟩ → ((𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋))) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
34	15, 33	syl6bi 253	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖∈_𝑔𝑗) → ((𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋))) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗))))
35	34	imp 408	. . . . . . . . . . 11 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈_𝑔𝑗)) → ((𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋))) ↔ (𝑎‘𝑖)𝐸(𝑎‘𝑗)))
36	35	rabbidv 3441	. . . . . . . . . 10 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈_𝑔𝑗)) → {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})
37	13, 36	jca 513	. . . . . . . . 9 ⊢ (((𝑖 ∈ ω ∧ 𝑗 ∈ ω) ∧ 𝑋 = (𝑖∈_𝑔𝑗)) → (𝑋 = (𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
38	37	ex 414	. . . . . . . 8 ⊢ ((𝑖 ∈ ω ∧ 𝑗 ∈ ω) → (𝑋 = (𝑖∈_𝑔𝑗) → (𝑋 = (𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
39	38	reximdva 3169	. . . . . . 7 ⊢ (𝑖 ∈ ω → (∃𝑗 ∈ ω 𝑋 = (𝑖∈_𝑔𝑗) → ∃𝑗 ∈ ω (𝑋 = (𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
40	39	reximia 3082	. . . . . 6 ⊢ (∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑋 = (𝑖∈_𝑔𝑗) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
41	12, 40	simplbiim 506	. . . . 5 ⊢ (𝑋 ∈ (Fmla‘∅) → ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑋 = (𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
43		simp3 1139	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → 𝑋 ∈ (Fmla‘∅))
44		ovex 7442	. . . . . 6 ⊢ (𝑀 ↑_m ω) ∈ V
45	44	rabex 5333	. . . . 5 ⊢ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} ∈ V
46		eqeq1 2737	. . . . . . . 8 ⊢ (𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} → (𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)} ↔ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}))
47	9, 46	bi2anan9 638	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))}) → ((𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)}) ↔ (𝑋 = (𝑖∈_𝑔𝑗) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
48	47	2rexbidv 3220	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})))
49	48	opelopabga 5534	. . . . 5 ⊢ ((𝑋 ∈ (Fmla‘∅) ∧ {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))} ∈ V) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))}⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈_𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})})
52	2	satfv0 34349	. . . . 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 2820	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))}⟩ ∈ (𝑆‘∅) ↔ ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))}⟩ ∈ (𝑆‘∅) ↔ ⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘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‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))}⟩ ∈ (𝑆‘∅))
56		funopfv 6944	. 2 ⊢ (Fun (𝑆‘∅) → (⟨𝑋, {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))}⟩ ∈ (𝑆‘∅) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))}))
57	6, 55, 56	sylc 65	1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑋 ∈ (Fmla‘∅)) → ((𝑆‘∅)‘𝑋) = {𝑎 ∈ (𝑀 ↑_m ω) ∣ (𝑎‘(1^st ‘(2^nd ‘𝑋)))𝐸(𝑎‘(2^nd ‘(2^nd ‘𝑋)))})

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 {crab 3433 Vcvv 3475 ∅c0 4323 ⟨cop 4635 class class class wbr 5149 {copab 5211 Fun wfun 6538 ‘cfv 6544 (class class class)co 7409 ωcom 7855 1^st c1st 7973 2^nd c2nd 7974 ↑_m cmap 8820 ∈_𝑔cgoe 34324 Sat csat 34327 Fmlacfmla 34328

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-map 8822 df-goel 34331 df-sat 34334 df-fmla 34336

This theorem is referenced by: satefvfmla0 34409

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