Theorem sprsymrelf1 47370

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 47369	. 2 ⊢ 𝐹:𝑃⟶𝑅
5	1, 2, 3	sprsymrelfv 47368	. . . . 5 ⊢ (𝑎 ∈ 𝑃 → (𝐹‘𝑎) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑎 𝑐 = {𝑥, 𝑦}})
6	1, 2, 3	sprsymrelfv 47368	. . . . 5 ⊢ (𝑏 ∈ 𝑃 → (𝐹‘𝑏) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑏 𝑐 = {𝑥, 𝑦}})
7	5, 6	eqeqan12d 2754	. . . 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 2836	. . . . . 6 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ∈ 𝒫 (Pairs‘𝑉))
9		vex 3492	. . . . . . 7 ⊢ 𝑎 ∈ V
10	9	elpw 4626	. . . . . 6 ⊢ (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
11	8, 10	bitri 275	. . . . 5 ⊢ (𝑎 ∈ 𝑃 ↔ 𝑎 ⊆ (Pairs‘𝑉))
12	1	eleq2i 2836	. . . . . 6 ⊢ (𝑏 ∈ 𝑃 ↔ 𝑏 ∈ 𝒫 (Pairs‘𝑉))
13		vex 3492	. . . . . . 7 ⊢ 𝑏 ∈ V
14	13	elpw 4626	. . . . . 6 ⊢ (𝑏 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑏 ⊆ (Pairs‘𝑉))
15	12, 14	bitri 275	. . . . 5 ⊢ (𝑏 ∈ 𝑃 ↔ 𝑏 ⊆ (Pairs‘𝑉))
16		sprsymrelf1lem 47365	. . . . . . . 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 406	. . . . . . 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 2747	. . . . . . . . . 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 47365	. . . . . . . . . 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	biimtrid 242	. . . . . . . . 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 458	. . . . . . . 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 406	. . . . . . 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 4026	. . . . . 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 412	. . . . 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 597	. . . 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 240	. . 3 ⊢ ((𝑎 ∈ 𝑃 ∧ 𝑏 ∈ 𝑃) → ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏))
27	26	rgen2 3205	. 2 ⊢ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)
28		dff13 7292	. 2 ⊢ (𝐹:𝑃–1-1→𝑅 ↔ (𝐹:𝑃⟶𝑅 ∧ ∀𝑎 ∈ 𝑃 ∀𝑏 ∈ 𝑃 ((𝐹‘𝑎) = (𝐹‘𝑏) → 𝑎 = 𝑏)))
29	4, 27, 28	mpbir2an 710	1 ⊢ 𝐹:𝑃–1-1→𝑅

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  {crab 3443   ⊆ wss 3976  𝒫 cpw 4622  {cpr 4650   class class class wbr 5166  {copab 5228   ↦ cmpt 5249   × cxp 5698  ⟶wf 6569  –1-1→wf1 6570  ‘cfv 6573  Pairscspr 47351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fv 6581  df-spr 47352

This theorem is referenced by:  sprsymrelf1o  47372