Theorem sprsymrelf1lem 48298

Description: Lemma for sprsymrelf1 48303. (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 48292	. . . . . 6 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑝 ∈ 𝑎) → ∃𝑖 ∈ 𝑉 ∃𝑗 ∈ 𝑉 𝑝 = {𝑖, 𝑗})
2	1	ad4ant14 765	. . . . 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 490	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → 𝑝 = {𝑖, 𝑗})
4	3	adantr 486	. . . . . . . . . . 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 2850	. . . . . . . . . 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 490	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} ∈ 𝑎)
7		eqeq1 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
8	7	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 = {𝑖, 𝑗} ↔ {𝑖, 𝑗} = {𝑖, 𝑗}))
9		eqidd 2766	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} = {𝑖, 𝑗})
10	6, 8, 9	rspcedvd 3585	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗})
11	10	adantlr 728	. . . . . . . . . . . 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 4703	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → {𝑥, 𝑦} = {𝑖, 𝑗})
13	12	eqeq2d 2776	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑖, 𝑗}))
14	13	rexbidv 3191	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗}))
15	14	opelopabga 5519	. . . . . . . . . . . . . 14 ⊢ ((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗}))
16	15	bicomd 226	. . . . . . . . . . . . 13 ⊢ ((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) → (∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗} ↔ ⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}}))
17	16	ad3antrrr 743	. . . . . . . . . . . 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 235	. . . . . . . . . . 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 418	. . . . . . . . . 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 243	. . . . . . . . 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 2854	. . . . . . . . . . 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 742	. . . . . . . . . 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 3191	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗}))
24	23	opelopabga 5519	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ V ∧ 𝑗 ∈ V) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗}))
25	24	el2v 3464	. . . . . . . . . . 11 ⊢ (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗})
26		eqtr3 2787	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑝 = 𝑐)
27	26	equcomd 2052	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑐 = 𝑝)
28	27	eleq1d 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 ∈ 𝑏 ↔ 𝑝 ∈ 𝑏))
29	28	biimpd 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → (𝑐 ∈ 𝑏 → 𝑝 ∈ 𝑏))
30	29	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 = {𝑖, 𝑗} → (𝑐 = {𝑖, 𝑗} → (𝑐 ∈ 𝑏 → 𝑝 ∈ 𝑏)))
31	30	com13 89	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ 𝑏 → (𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏)))
32	31	imp 412	. . . . . . . . . . . . . . 15 ⊢ ((𝑐 ∈ 𝑏 ∧ 𝑐 = {𝑖, 𝑗}) → (𝑝 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
33	32	rexlimiva 3160	. . . . . . . . . . . . . 14 ⊢ (∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗} → (𝑝 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
34	33	com12 33	. . . . . . . . . . . . 13 ⊢ (𝑝 = {𝑖, 𝑗} → (∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
35	34	adantl 487	. . . . . . . . . . . 12 ⊢ (((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) → (∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗} → 𝑝 ∈ 𝑏))
36	35	adantr 486	. . . . . . . . . . 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 245	. . . . . . . . . 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 243	. . . . . . . . 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 48	. . . . . . . 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 459	. . . . . . 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 3170	. . . . . 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 3161	. . . . 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 39	. . . 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 418	. . 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 3944	. 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 418	1 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  Vcvv 3457   ⊆ wss 3906  {cpr 4593  ⟨cop 4597  {copab 5175  ‘cfv 6540  Pairscspr 48284

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-spr 48285

This theorem is used by:  sprsymrelf1  48303