Theorem sprsymrelf1lem 47489

Description: Lemma for sprsymrelf1 47494. (Contributed by AV, 22-Nov-2021.)

Assertion

Ref	Expression
sprsymrelf1lem	⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏))

Distinct variable groups:   𝑉,𝑐   𝑎,𝑏,𝑐,𝑥,𝑦

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑎,𝑏)

Proof of Theorem sprsymrelf1lem

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

Step	Hyp	Ref	Expression
1		prssspr 47483	. . . . . 6 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑝 ∈ 𝑎) → ∃𝑖 ∈ 𝑉 ∃𝑗 ∈ 𝑉 𝑝 = {𝑖, 𝑗})
2	1	ad4ant14 752	. . . . 5 ⊢ ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝 ∈ 𝑎) → ∃𝑖 ∈ 𝑉 ∃𝑗 ∈ 𝑉 𝑝 = {𝑖, 𝑗})
3		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 = {𝑖, 𝑗})
4	3	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → 𝑝 = {𝑖, 𝑗})
5	4	eleq1d 2813	. . . . . . . . . 10 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → (𝑝 ∈ 𝑎 ↔ {𝑖, 𝑗} ∈ 𝑎))
6		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} ∈ 𝑎)
7		eqeq1 2733	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
8	7	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
9		eqidd 2730	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} = {𝑖, 𝑗})
10	6, 8, 9	rspcedvd 3590	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗})
11	10	adantlr 715	. . . . . . . . . . . 12 ⊢ (((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗})
12		preq12 4699	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → {𝑥, 𝑦} = {𝑖, 𝑗})
13	12	eqeq2d 2740	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑖, 𝑗}))
14	13	rexbidv 3157	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗}))
15	14	opelopabga 5493	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗}))
16	15	bicomd 223	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) → (∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}}))
17	16	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) ∧ {𝑖, 𝑗} ∈ 𝑎) → (∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}}))
18	11, 17	mpbid 232	. . . . . . . . . . 11 ⊢ (((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) ∧ {𝑖, 𝑗} ∈ 𝑎) → ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}})
19	18	ex 412	. . . . . . . . . 10 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → ({𝑖, 𝑗} ∈ 𝑎 → ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}}))
20	5, 19	sylbid 240	. . . . . . . . 9 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → (𝑝 ∈ 𝑎 → ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}}))
21		eleq2 2817	. . . . . . . . . . 11 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}))
22	21	ad2antll 729	. . . . . . . . . 10 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}))
23	13	rexbidv 3157	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗}))
24	23	opelopabga 5493	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ V ∧ 𝑗 ∈ V) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗}))
25	24	el2v 3454	. . . . . . . . . . 11 ⊢ (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗})
26		eqtr3 2751	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑝 = 𝑐)
27	26	equcomd 2019	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑐 = 𝑝)
28	27	eleq1d 2813	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 ∈ 𝑏 ↔ 𝑝 ∈ 𝑏))
29	28	biimpd 229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 ∈ 𝑏 → 𝑝 ∈ 𝑏))
30	29	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} → (𝑐 ∈ 𝑏 → 𝑝 ∈ 𝑏)))
31	30	com13 88	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝑏 → (𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏)))
32	31	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝑏 ∧ 𝑐 = {𝑖, 𝑗}) → (𝑝 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
33	32	rexlimiva 3126	. . . . . . . . . . . . . 14 ⊢ (∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
34	33	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑝 = {𝑖, 𝑗} → (∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
35	34	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → (∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
36	35	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → (∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
37	25, 36	biimtrid 242	. . . . . . . . . 10 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑝 ∈ 𝑏))
38	22, 37	sylbid 240	. . . . . . . . 9 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} → 𝑝 ∈ 𝑏))
39	20, 38	syld 47	. . . . . . . 8 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})) → (𝑝 ∈ 𝑎 → 𝑝 ∈ 𝑏))
40	39	expimpd 453	. . . . . . 7 ⊢ (((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑏))
41	40	rexlimdva2 3136	. . . . . 6 ⊢ (𝑖 ∈ 𝑉 → (∃𝑗 ∈ 𝑉 𝑝 = {𝑖, 𝑗} → ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑏)))
42	41	rexlimiv 3127	. . . . 5 ⊢ (∃𝑖 ∈ 𝑉 ∃𝑗 ∈ 𝑉 𝑝 = {𝑖, 𝑗} → ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑏))
43	2, 42	mpcom 38	. . . 4 ⊢ ((((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝑏)
44	43	ex 412	. . 3 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) → (𝑝 ∈ 𝑎 → 𝑝 ∈ 𝑏))
45	44	ssrdv 3952	. 2 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎 ⊆ 𝑏)
46	45	ex 412	1 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  {cpr 4591  ⟨cop 4595  {copab 5169  ‘cfv 6511  Pairscspr 47475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-spr 47476

This theorem is referenced by:  sprsymrelf1  47494