Theorem sprsymrelf1lem 47851

Description: Lemma for sprsymrelf1 47856. (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 47845	. . . . . 6 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑝 ∈ 𝑎) → ∃𝑖 ∈ 𝑉 ∃𝑗 ∈ 𝑉 𝑝 = {𝑖, 𝑗})
2	1	ad4ant14 753	. . . . 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 2822	. . . . . . . . . 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 2741	. . . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → {𝑖, 𝑗} = {𝑖, 𝑗})
10	6, 8, 9	rspcedvd 3580	. . . . . . . . . . . . 13 ⊢ ((((𝑖 ∈ 𝑉 ∧ 𝑗 ∈ 𝑉) ∧ 𝑝 = {𝑖, 𝑗}) ∧ {𝑖, 𝑗} ∈ 𝑎) → ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗})
11	10	adantlr 716	. . . . . . . . . . . 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 4694	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → {𝑥, 𝑦} = {𝑖, 𝑗})
13	12	eqeq2d 2748	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (𝑐 = {𝑥, 𝑦} ↔ 𝑐 = {𝑖, 𝑗}))
14	13	rexbidv 3162	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑎 𝑐 = {𝑖, 𝑗}))
15	14	opelopabga 5489	. . . . . . . . . . . . . 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 731	. . . . . . . . . . . 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 2826	. . . . . . . . . . 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 730	. . . . . . . . . 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 3162	. . . . . . . . . . . . 13 ⊢ ((𝑥 = 𝑖 ∧ 𝑦 = 𝑗) → (∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗}))
24	23	opelopabga 5489	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ V ∧ 𝑗 ∈ V) → (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗}))
25	24	el2v 3449	. . . . . . . . . . 11 ⊢ (⟨𝑖, 𝑗⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ ∃𝑐 ∈ 𝑏 𝑐 = {𝑖, 𝑗})
26		eqtr3 2759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑝 = 𝑐)
27	26	equcomd 2021	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 = {𝑖, 𝑗} ∧ 𝑐 = {𝑖, 𝑗}) → 𝑐 = 𝑝)
28	27	eleq1d 2822	. . . . . . . . . . . . . . . . . . 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 3131	. . . . . . . . . . . . . 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 3141	. . . . . 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 3132	. . . . 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 3941	. 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 1542   ∈ wcel 2114  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  {cpr 4584  ⟨cop 4588  {copab 5162  ‘cfv 6500  Pairscspr 47837

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 5226  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  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 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-iota 6456  df-fun 6502  df-fv 6508  df-spr 47838

This theorem is referenced by:  sprsymrelf1  47856