Theorem sprsymrelf1 44948

Description: The mapping 𝐹 is a one-to-one function from the subsets of the set of pairs over a fixed set 𝑉 into the symmetric relations 𝑅 on the fixed set 𝑉. (Contributed by AV, 19-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
sprsymrelf1	⊢ 𝐹:𝑃–1-1→𝑅

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

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

Proof of Theorem sprsymrelf1

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

Step	Hyp	Ref	Expression
1		sprsymrelf.p	. . 3 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
2		sprsymrelf.r	. . 3 ⊢ 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (𝑥𝑟𝑦 ↔ 𝑦𝑟𝑥)}
3		sprsymrelf.f	. . 3 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
4	1, 2, 3	sprsymrelf 44947	. 2 ⊢ 𝐹:𝑃⟶𝑅
5	1, 2, 3	sprsymrelfv 44946	. . . . 5 ⊢ (𝑎 ∈ 𝑃 → (𝐹‘𝑎) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}})
6	1, 2, 3	sprsymrelfv 44946	. . . . 5 ⊢ (𝑏 ∈ 𝑃 → (𝐹‘𝑏) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})
7	5, 6	eqeqan12d 2752	. . . 4 ⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → ((𝐹‘𝑎) = (𝐹‘𝑏) ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}))
8	1	eleq2i 2830	. . . . . 6 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ∈ 𝒫 (Pairs‘𝑉))
9		vex 3436	. . . . . . 7 ⊢ 𝑎 ∈ V
10	9	elpw 4537	. . . . . 6 ⊢ (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
11	8, 10	bitri 274	. . . . 5 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ⊆ (Pairs‘𝑉))
12	1	eleq2i 2830	. . . . . 6 ⊢ (𝑏 ∈ 𝑃 ↔ 𝑏 ∈ 𝒫 (Pairs‘𝑉))
13		vex 3436	. . . . . . 7 ⊢ 𝑏 ∈ V
14	13	elpw 4537	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑏 ⊆ (Pairs‘𝑉))
15	12, 14	bitri 274	. . . . 5 ⊢ (𝑏 ∈ 𝑃 ↔ 𝑏 ⊆ (Pairs‘𝑉))
16		sprsymrelf1lem 44943	. . . . . . . 8 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 ⊆ 𝑏))
17	16	imp 407	. . . . . . 7 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎 ⊆ 𝑏)
18		eqcom 2745	. . . . . . . . . 10 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}})
19		sprsymrelf1lem 44943	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ (Pairs‘𝑉) ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} → 𝑏 ⊆ 𝑎))
20	18, 19	syl5bi 241	. . . . . . . . 9 ⊢ ((𝑏 ⊆ (Pairs‘𝑉) ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑏 ⊆ 𝑎))
21	20	ancoms 459	. . . . . . . 8 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑏 ⊆ 𝑎))
22	21	imp 407	. . . . . . 7 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑏 ⊆ 𝑎)
23	17, 22	eqssd 3938	. . . . . 6 ⊢ (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎 = 𝑏)
24	23	ex 413	. . . . 5 ⊢ ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 = 𝑏))
25	11, 15, 24	syl2anb 598	. . . 4 ⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 = 𝑏))
26	7, 25	sylbid 239	. . 3 ⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
27	26	rgen2 3120	. 2 ⊢ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
28		dff13 7128	. 2 ⊢ (𝐹:𝑃–1-1→𝑅 ↔ (𝐹:𝑃⟶𝑅 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
29	4, 27, 28	mpbir2an 708	1 ⊢ 𝐹:𝑃–1-1→𝑅

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068   ⊆ wss 3887  𝒫 cpw 4533  {cpr 4563   class class class wbr 5074  {copab 5136   ↦ cmpt 5157   × cxp 5587  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433  Pairscspr 44929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441  df-spr 44930

This theorem is referenced by:  sprsymrelf1o  44950