Theorem sprsymrelfo 44837

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 44835	. . 3 ⊢ 𝐹:𝑃⟶𝑅
5	4	a1i 11	. 2 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃⟶𝑅)
6		breq 5072	. . . . . . . . 9 ⊢ (𝑟 = 𝑡 → (𝑥𝑟𝑦 ↔ 𝑥𝑡𝑦))
7		breq 5072	. . . . . . . . 9 ⊢ (𝑟 = 𝑡 → (𝑦𝑟𝑥 ↔ 𝑦𝑡𝑥))
8	6, 7	bibi12d 345	. . . . . . . 8 ⊢ (𝑟 = 𝑡 → ((𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)))
9	8	2ralbidv 3122	. . . . . . 7 ⊢ (𝑟 = 𝑡 → (∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥) ↔ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)))
10	9, 2	elrab2 3620	. . . . . 6 ⊢ (𝑡 ∈ 𝑅 ↔ (𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)))
11		eqid 2738	. . . . . . . . . . 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 44832	. . . . . . . . . 10 ⊢ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝒫 (Pairs‘𝑉)
13	12, 1	eleqtrri 2838	. . . . . . . . 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 3334	. . . . . . . . . . 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 5136	. . . . . . . . . 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 2749	. . . . . . . . 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 481	. . . . . . . 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 4535	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝒫 (𝑉 × 𝑉) ↔ 𝑡 ⊆ (𝑉 × 𝑉))
20		xpss 5596	. . . . . . . . . . . . . . . 16 ⊢ (𝑉 × 𝑉) ⊆ (V × V)
21		sstr2 3924	. . . . . . . . . . . . . . . 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 5587	. . . . . . . . . . . . . . 15 ⊢ (Rel 𝑡 ↔ 𝑡 ⊆ (V × V))
24	22, 23	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (𝑡 ⊆ (𝑉 × 𝑉) → Rel 𝑡)
25	24	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) → Rel 𝑡)
26		dfrel4v 6082	. . . . . . . . . . . . . 14 ⊢ (Rel 𝑡 ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦})
27		nfv 1918	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥(𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉))
28		nfra1 3142	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)
29	27, 28	nfan 1903	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥))
30		nfv 1918	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦(𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉))
31		nfra2w 3151	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥)
32	30, 31	nfan 1903	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑦((𝑉 ∈ 𝑊 ∧ 𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑡𝑦 ↔ 𝑦𝑡𝑥))
33	11	sprsymrelfolem2 44833	. . . . . . . . . . . . . . . . . . . 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 1116	. . . . . . . . . . . . . . . . . . 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 5135	. . . . . . . . . . . . . . . . . 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 2749	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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	syl5bi 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 413	. . . . . . . . . . 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 417	. . . . . . . 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 3555	. . . . . . 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 412	. . . . . 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 407	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅) → ∃𝑓 ∈ 𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}})
50	1, 2, 3	sprsymrelfv 44834	. . . . . . 7 ⊢ (𝑓 ∈ 𝑃 → (𝐹‘𝑓) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}})
51	50	adantl 481	. . . . . 6 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅) ∧ 𝑓 ∈ 𝑃) → (𝐹‘𝑓) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}})
52	51	eqeq2d 2749	. . . . 5 ⊢ (((𝑉 ∈ 𝑊 ∧ 𝑡 ∈ 𝑅) ∧ 𝑓 ∈ 𝑃) → (𝑡 = (𝐹‘𝑓) ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑓 𝑐 = {𝑥, 𝑦}}))
53	52	rexbidva 3224	. . . 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 3107	. 2 ⊢ (𝑉 ∈ 𝑊 → ∀𝑡 ∈ 𝑅 ∃𝑓 ∈ 𝑃 𝑡 = (𝐹‘𝑓))
56		dffo3 6960	. 2 ⊢ (𝐹:𝑃–onto→𝑅 ↔ (𝐹:𝑃⟶𝑅 ∧ ∀𝑡 ∈ 𝑅 ∃𝑓 ∈ 𝑃 𝑡 = (𝐹‘𝑓)))
57	5, 55, 56	sylanbrc 582	1 ⊢ (𝑉 ∈ 𝑊 → 𝐹:𝑃–onto→𝑅)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  {crab 3067  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  {cpr 4560   class class class wbr 5070  {copab 5132   ↦ cmpt 5153   × cxp 5578  Rel wrel 5585  ⟶wf 6414  –onto→wfo 6416  ‘cfv 6418  Pairscspr 44817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-spr 44818

This theorem is referenced by:  sprsymrelf1o  44838