Theorem sprsymrelfo 45679

Description: The mapping 𝐹 is a function from the subsets of the set of pairs over a fixed set 𝑉 onto the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 23-Nov-2021.)

Hypotheses

Ref	Expression
sprsymrelf.p	⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
sprsymrelf.r	⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
sprsymrelf.f	⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})

Assertion

Ref	Expression
sprsymrelfo	⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅)

Distinct variable groups:   𝑃,𝑝   𝑉,𝑐,𝑥,𝑦   𝑝,𝑐,𝑥,𝑦,𝑟   𝑅,𝑝   𝑉,𝑟,𝑐,𝑥,𝑦   𝑊,𝑐,𝑥,𝑦

Allowed substitution hints:   𝑃(𝑥,𝑦,𝑟,𝑐)   𝑅(𝑥,𝑦,𝑟,𝑐)   𝐹(𝑥,𝑦,𝑟,𝑝,𝑐)   𝑉(𝑝)   𝑊(𝑟,𝑝)

Proof of Theorem sprsymrelfo

Dummy variables 𝑎 𝑏 𝑓 𝑞 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sprsymrelf.p	. . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
2		sprsymrelf.r	. . . 4 ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
3		sprsymrelf.f	. . . 4 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
4	1, 2, 3	sprsymrelf 45677	. . 3 ⊢ 𝐹:𝑃⟶𝑅
5	4	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃⟶𝑅)
6		breq 5107	. . . . . . . . 9 ⊢ (𝑟 = 𝑡 → (𝑥𝑟𝑦 ↔ 𝑥𝑡𝑦))
7		breq 5107	. . . . . . . . 9 ⊢ (𝑟 = 𝑡 → (𝑦𝑟𝑥 ↔ 𝑦𝑡𝑥))
8	6, 7	bibi12d 345	. . . . . . . 8 ⊢ (𝑟 = 𝑡 → ((𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)))
9	8	2ralbidv 3212	. . . . . . 7 ⊢ (𝑟 = 𝑡 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)))
10	9, 2	elrab2 3648	. . . . . 6 ⊢ (𝑡 ∈ 𝑅 ↔ (𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)))
11		eqid 2736	. . . . . . . . . . 11 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}
12	11	sprsymrelfolem1 45674	. . . . . . . . . 10 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝒫 (Pairs‘𝑉)
13	12, 1	eleqtrri 2837	. . . . . . . . 9 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝑃
14	13	a1i 11	. . . . . . . 8 ⊢ (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) ∧ 𝑉 ∈ 𝑊) → {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝑃)
15		rexeq 3310	. . . . . . . . . . 11 ⊢ (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → (∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
16	15	opabbidv 5171	. . . . . . . . . 10 ⊢ (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
17	16	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
18	17	adantl 482	. . . . . . . 8 ⊢ ((((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) ∧ 𝑉 ∈ 𝑊) ∧ 𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
19		velpw 4565	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝒫 (𝑉 × 𝑉) ↔ 𝑡 ⊆ (𝑉 × 𝑉))
20		xpss 5649	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 × 𝑉) ⊆ (V × V)
21		sstr2 3951	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ⊆ (𝑉 × 𝑉) → ((𝑉 × 𝑉) ⊆ (V × V) → 𝑡 ⊆ (V × V)))
22	20, 21	mpi 20	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ⊆ (𝑉 × 𝑉) → 𝑡 ⊆ (V × V))
23		df-rel 5640	. . . . . . . . . . . . . . 15 ⊢ (Rel 𝑡 ↔ 𝑡 ⊆ (V × V))
24	22, 23	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (𝑡 ⊆ (𝑉 × 𝑉) → Rel 𝑡)
25	24	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) → Rel 𝑡)
26		dfrel4v 6142	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑡 ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦})
27		nfv 1917	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉))
28		nfra1 3267	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)
29	27, 28	nfan 1902	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥))
30		nfv 1917	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦(𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉))
31		nfra2w 3282	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)
32	30, 31	nfan 1902	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥))
33	11	sprsymrelfolem2 45675	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) → (𝑥𝑡𝑦 ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
34	33	3expa 1118	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) → (𝑥𝑡𝑦 ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
35	29, 32, 34	opabbid 5170	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
36	35	eqeq2d 2747	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
37	36	biimpd 228	. . . . . . . . . . . . . . . 16 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
38	37	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
39	38	com23 86	. . . . . . . . . . . . . 14 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
40	26, 39	biimtrid 241	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) → (Rel 𝑡 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
41	25, 40	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
42	41	expcom 414	. . . . . . . . . . 11 ⊢ (𝑡 ⊆ (𝑉 × 𝑉) → (𝑉 ∈ 𝑊 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
43	42	com23 86	. . . . . . . . . 10 ⊢ (𝑡 ⊆ (𝑉 × 𝑉) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥) → (𝑉 ∈ 𝑊 → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
44	19, 43	sylbi 216	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝒫 (𝑉 × 𝑉) → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥) → (𝑉 ∈ 𝑊 → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
45	44	imp31 418	. . . . . . . 8 ⊢ (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) ∧ 𝑉 ∈ 𝑊) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
46	14, 18, 45	rspcedvd 3583	. . . . . . 7 ⊢ (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) ∧ 𝑉 ∈ 𝑊) → ∃𝑓 ∈ 𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}})
47	46	ex 413	. . . . . 6 ⊢ ((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)) → (𝑉 ∈ 𝑊 → ∃𝑓 ∈ 𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}}))
48	10, 47	sylbi 216	. . . . 5 ⊢ (𝑡 ∈ 𝑅 → (𝑉 ∈ 𝑊 → ∃𝑓 ∈ 𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}}))
49	48	impcom 408	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅) → ∃𝑓 ∈ 𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}})
50	1, 2, 3	sprsymrelfv 45676	. . . . . . 7 ⊢ (𝑓 ∈ 𝑃 → (𝐹‘𝑓) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}})
51	50	adantl 482	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅) ∧ 𝑓 ∈ 𝑃) → (𝐹‘𝑓) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}})
52	51	eqeq2d 2747	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅) ∧ 𝑓 ∈ 𝑃) → (𝑡 = (𝐹‘𝑓) ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}}))
53	52	rexbidva 3173	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅) → (∃𝑓 ∈ 𝑃 𝑡 = (𝐹‘𝑓) ↔ ∃𝑓 ∈ 𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}}))
54	49, 53	mpbird 256	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅) → ∃𝑓 ∈ 𝑃 𝑡 = (𝐹‘𝑓))
55	54	ralrimiva 3143	. 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑡 ∈ 𝑅 ∃𝑓 ∈ 𝑃 𝑡 = (𝐹‘𝑓))
56		dffo3 7052	. 2 ⊢ (𝐹:𝑃–onto→𝑅 ↔ (𝐹:𝑃⟶𝑅 ∧ ∀𝑡 ∈ 𝑅 ∃𝑓 ∈ 𝑃 𝑡 = (𝐹‘𝑓)))
57	5, 55, 56	sylanbrc 583	1 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3064  ∃wrex 3073  {crab 3407  Vcvv 3445   ⊆ wss 3910  𝒫 cpw 4560  {cpr 4588   class class class wbr 5105  {copab 5167   ↦ cmpt 5188   × cxp 5631  Rel wrel 5638  ⟶wf 6492  –onto→wfo 6494  ‘cfv 6496  Pairscspr 45659

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-spr 45660

This theorem is referenced by:  sprsymrelf1o  45680