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Theorem sprsymrelfo 42072
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) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
sprsymrelf.f 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
Assertion
Ref Expression
sprsymrelfo (𝑉𝑊𝐹:𝑃onto𝑅)
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.
StepHypRef Expression
1 sprsymrelf.p . . . 4 𝑃 = 𝒫 (Pairs‘𝑉)
2 sprsymrelf.r . . . 4 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
3 sprsymrelf.f . . . 4 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
41, 2, 3sprsymrelf 42070 . . 3 𝐹:𝑃𝑅
54a1i 11 . 2 (𝑉𝑊𝐹:𝑃𝑅)
6 breq 4687 . . . . . . . . 9 (𝑟 = 𝑡 → (𝑥𝑟𝑦𝑥𝑡𝑦))
7 breq 4687 . . . . . . . . 9 (𝑟 = 𝑡 → (𝑦𝑟𝑥𝑦𝑡𝑥))
86, 7bibi12d 334 . . . . . . . 8 (𝑟 = 𝑡 → ((𝑥𝑟𝑦𝑦𝑟𝑥) ↔ (𝑥𝑡𝑦𝑦𝑡𝑥)))
982ralbidv 3018 . . . . . . 7 (𝑟 = 𝑡 → (∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥) ↔ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)))
109, 2elrab2 3399 . . . . . 6 (𝑡𝑅 ↔ (𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)))
11 eqid 2651 . . . . . . . . . . 11 {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}
1211sprsymrelfolem1 42067 . . . . . . . . . 10 {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝒫 (Pairs‘𝑉)
1312, 1eleqtrri 2729 . . . . . . . . 9 {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝑃
1413a1i 11 . . . . . . . 8 (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) ∧ 𝑉𝑊) → {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} ∈ 𝑃)
15 rexeq 3169 . . . . . . . . . . 11 (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → (∃𝑐𝑓 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
1615opabbidv 4749 . . . . . . . . . 10 (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
1716eqeq2d 2661 . . . . . . . . 9 (𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)} → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
1817adantl 481 . . . . . . . 8 ((((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) ∧ 𝑉𝑊) ∧ 𝑓 = {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
19 selpw 4198 . . . . . . . . . 10 (𝑡 ∈ 𝒫 (𝑉 × 𝑉) ↔ 𝑡 ⊆ (𝑉 × 𝑉))
20 xpss 5159 . . . . . . . . . . . . . . . 16 (𝑉 × 𝑉) ⊆ (V × V)
21 sstr2 3643 . . . . . . . . . . . . . . . 16 (𝑡 ⊆ (𝑉 × 𝑉) → ((𝑉 × 𝑉) ⊆ (V × V) → 𝑡 ⊆ (V × V)))
2220, 21mpi 20 . . . . . . . . . . . . . . 15 (𝑡 ⊆ (𝑉 × 𝑉) → 𝑡 ⊆ (V × V))
23 df-rel 5150 . . . . . . . . . . . . . . 15 (Rel 𝑡𝑡 ⊆ (V × V))
2422, 23sylibr 224 . . . . . . . . . . . . . 14 (𝑡 ⊆ (𝑉 × 𝑉) → Rel 𝑡)
2524adantl 481 . . . . . . . . . . . . 13 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → Rel 𝑡)
26 dfrel4v 5619 . . . . . . . . . . . . . 14 (Rel 𝑡𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦})
27 nfv 1883 . . . . . . . . . . . . . . . . . . . 20 𝑥(𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉))
28 nfra1 2970 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)
2927, 28nfan 1868 . . . . . . . . . . . . . . . . . . 19 𝑥((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥))
30 nfv 1883 . . . . . . . . . . . . . . . . . . . 20 𝑦(𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉))
31 nfra2 2975 . . . . . . . . . . . . . . . . . . . 20 𝑦𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)
3230, 31nfan 1868 . . . . . . . . . . . . . . . . . . 19 𝑦((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥))
3311sprsymrelfolem2 42068 . . . . . . . . . . . . . . . . . . . 20 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑥𝑡𝑦 ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
34333expa 1284 . . . . . . . . . . . . . . . . . . 19 (((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑥𝑡𝑦 ↔ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}))
3529, 32, 34opabbid 4748 . . . . . . . . . . . . . . . . . 18 (((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
3635eqeq2d 2661 . . . . . . . . . . . . . . . . 17 (((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
3736biimpd 219 . . . . . . . . . . . . . . . 16 (((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
3837ex 449 . . . . . . . . . . . . . . 15 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
3938com23 86 . . . . . . . . . . . . . 14 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → (𝑡 = {⟨𝑥, 𝑦⟩ ∣ 𝑥𝑡𝑦} → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4026, 39syl5bi 232 . . . . . . . . . . . . 13 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → (Rel 𝑡 → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4125, 40mpd 15 . . . . . . . . . . . 12 ((𝑉𝑊𝑡 ⊆ (𝑉 × 𝑉)) → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}}))
4241expcom 450 . . . . . . . . . . 11 (𝑡 ⊆ (𝑉 × 𝑉) → (𝑉𝑊 → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4342com23 86 . . . . . . . . . 10 (𝑡 ⊆ (𝑉 × 𝑉) → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → (𝑉𝑊𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4419, 43sylbi 207 . . . . . . . . 9 (𝑡 ∈ 𝒫 (𝑉 × 𝑉) → (∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥) → (𝑉𝑊𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})))
4544imp31 447 . . . . . . . 8 (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) ∧ 𝑉𝑊) → 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ {𝑞 ∈ (Pairs‘𝑉) ∣ ∀𝑎𝑉𝑏𝑉 (𝑞 = {𝑎, 𝑏} → 𝑎𝑡𝑏)}𝑐 = {𝑥, 𝑦}})
4614, 18, 45rspcedvd 3348 . . . . . . 7 (((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) ∧ 𝑉𝑊) → ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}})
4746ex 449 . . . . . 6 ((𝑡 ∈ 𝒫 (𝑉 × 𝑉) ∧ ∀𝑥𝑉𝑦𝑉 (𝑥𝑡𝑦𝑦𝑡𝑥)) → (𝑉𝑊 → ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}}))
4810, 47sylbi 207 . . . . 5 (𝑡𝑅 → (𝑉𝑊 → ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}}))
4948impcom 445 . . . 4 ((𝑉𝑊𝑡𝑅) → ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}})
501, 2, 3sprsymrelfv 42069 . . . . . . 7 (𝑓𝑃 → (𝐹𝑓) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}})
5150adantl 481 . . . . . 6 (((𝑉𝑊𝑡𝑅) ∧ 𝑓𝑃) → (𝐹𝑓) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}})
5251eqeq2d 2661 . . . . 5 (((𝑉𝑊𝑡𝑅) ∧ 𝑓𝑃) → (𝑡 = (𝐹𝑓) ↔ 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}}))
5352rexbidva 3078 . . . 4 ((𝑉𝑊𝑡𝑅) → (∃𝑓𝑃 𝑡 = (𝐹𝑓) ↔ ∃𝑓𝑃 𝑡 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑓 𝑐 = {𝑥, 𝑦}}))
5449, 53mpbird 247 . . 3 ((𝑉𝑊𝑡𝑅) → ∃𝑓𝑃 𝑡 = (𝐹𝑓))
5554ralrimiva 2995 . 2 (𝑉𝑊 → ∀𝑡𝑅𝑓𝑃 𝑡 = (𝐹𝑓))
56 dffo3 6414 . 2 (𝐹:𝑃onto𝑅 ↔ (𝐹:𝑃𝑅 ∧ ∀𝑡𝑅𝑓𝑃 𝑡 = (𝐹𝑓)))
575, 55, 56sylanbrc 699 1 (𝑉𝑊𝐹:𝑃onto𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  𝒫 cpw 4191  {cpr 4212   class class class wbr 4685  {copab 4745  cmpt 4762   × cxp 5141  Rel wrel 5148  wf 5922  ontowfo 5924  cfv 5926  Pairscspr 42052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-spr 42053
This theorem is referenced by:  sprsymrelf1o  42073
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