Theorem sprsymrelf1lem 48058

Description: Lemma for sprsymrelf1 48063. (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 48052	. . . . . 6 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑝 ∈ 𝑎) → ∃𝑖 ∈ 𝑉 ∃𝑗 ∈ 𝑉 𝑝 = {𝑖, 𝑗})
2	1	ad4ant14 762	. . . . 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 488	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 = {𝑖, 𝑗})
4	3	adantr 484	. . . . . . . . . . 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 2846	. . . . . . . . . 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 488	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} ∈ 𝑎)
7		eqeq1 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
8	7	adantl 485	. . . . . . . . . . . . . 14 ⊢ (((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
9		eqidd 2762	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} = {𝑖, 𝑗})
10	6, 8, 9	rspcedvd 3582	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗})
11	10	adantlr 725	. . . . . . . . . . . 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 4691	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → {𝑥, 𝑦} = {𝑖, 𝑗})
13	12	eqeq2d 2772	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑖, 𝑗}))
14	13	rexbidv 3185	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗}))
15	14	opelopabga 5500	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗}))
16	15	bicomd 225	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) → (∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}}))
17	16	ad3antrrr 740	. . . . . . . . . . . 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 234	. . . . . . . . . . 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 416	. . . . . . . . . 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 242	. . . . . . . . 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 2850	. . . . . . . . . . 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 739	. . . . . . . . . 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 3185	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗}))
24	23	opelopabga 5500	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ V ∧ 𝑗 ∈ V) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗}))
25	24	el2v 3460	. . . . . . . . . . 11 ⊢ (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗})
26		eqtr3 2783	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑝 = 𝑐)
27	26	equcomd 2038	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑐 = 𝑝)
28	27	eleq1d 2846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 ∈ 𝑏 ↔ 𝑝 ∈ 𝑏))
29	28	biimpd 231	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 ∈ 𝑏 → 𝑝 ∈ 𝑏))
30	29	ex 416	. . . . . . . . . . . . . . . . 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 410	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝑏 ∧ 𝑐 = {𝑖, 𝑗}) → (𝑝 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
33	32	rexlimiva 3154	. . . . . . . . . . . . . 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 485	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → (∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
36	35	adantr 484	. . . . . . . . . . 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 244	. . . . . . . . . 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 242	. . . . . . . . 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 457	. . . . . . 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 3164	. . . . . 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 3155	. . . . 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 416	. . 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 3940	. 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 416	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 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∃wrex 3085  Vcvv 3453   ⊆ wss 3902  {cpr 4581  ⟨cop 4585  {copab 5159  ‘cfv 6516  Pairscspr 48044

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6472  df-fun 6518  df-fv 6524  df-spr 48045

This theorem is referenced by:  sprsymrelf1  48063