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Theorem sprsymrelf1 42545
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) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
sprsymrelf.f 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
Assertion
Ref Expression
sprsymrelf1 𝐹:𝑃1-1𝑅
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.
StepHypRef Expression
1 sprsymrelf.p . . 3 𝑃 = 𝒫 (Pairs‘𝑉)
2 sprsymrelf.r . . 3 𝑅 = {𝑟 ∈ 𝒫 (𝑉 × 𝑉) ∣ ∀𝑥𝑉𝑦𝑉 (𝑥𝑟𝑦𝑦𝑟𝑥)}
3 sprsymrelf.f . . 3 𝐹 = (𝑝𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑝 𝑐 = {𝑥, 𝑦}})
41, 2, 3sprsymrelf 42544 . 2 𝐹:𝑃𝑅
51, 2, 3sprsymrelfv 42543 . . . . 5 (𝑎𝑃 → (𝐹𝑎) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}})
61, 2, 3sprsymrelfv 42543 . . . . 5 (𝑏𝑃 → (𝐹𝑏) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}})
75, 6eqeqan12d 2815 . . . 4 ((𝑎𝑃𝑏𝑃) → ((𝐹𝑎) = (𝐹𝑏) ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}))
81eleq2i 2870 . . . . . 6 (𝑎𝑃𝑎 ∈ 𝒫 (Pairs‘𝑉))
9 vex 3388 . . . . . . 7 𝑎 ∈ V
109elpw 4355 . . . . . 6 (𝑎 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑎 ⊆ (Pairs‘𝑉))
118, 10bitri 267 . . . . 5 (𝑎𝑃𝑎 ⊆ (Pairs‘𝑉))
121eleq2i 2870 . . . . . 6 (𝑏𝑃𝑏 ∈ 𝒫 (Pairs‘𝑉))
13 vex 3388 . . . . . . 7 𝑏 ∈ V
1413elpw 4355 . . . . . 6 (𝑏 ∈ 𝒫 (Pairs‘𝑉) ↔ 𝑏 ⊆ (Pairs‘𝑉))
1512, 14bitri 267 . . . . 5 (𝑏𝑃𝑏 ⊆ (Pairs‘𝑉))
16 sprsymrelf1lem 42540 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎𝑏))
1716imp 396 . . . . . . 7 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎𝑏)
18 eqcom 2806 . . . . . . . . . 10 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} ↔ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}})
19 sprsymrelf1lem 42540 . . . . . . . . . 10 ((𝑏 ⊆ (Pairs‘𝑉) ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} → 𝑏𝑎))
2018, 19syl5bi 234 . . . . . . . . 9 ((𝑏 ⊆ (Pairs‘𝑉) ∧ 𝑎 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑏𝑎))
2120ancoms 451 . . . . . . . 8 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑏𝑎))
2221imp 396 . . . . . . 7 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑏𝑎)
2317, 22eqssd 3815 . . . . . 6 (((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) ∧ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}}) → 𝑎 = 𝑏)
2423ex 402 . . . . 5 ((𝑎 ⊆ (Pairs‘𝑉) ∧ 𝑏 ⊆ (Pairs‘𝑉)) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 = 𝑏))
2511, 15, 24syl2anb 592 . . . 4 ((𝑎𝑃𝑏𝑃) → ({⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑎 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐𝑏 𝑐 = {𝑥, 𝑦}} → 𝑎 = 𝑏))
267, 25sylbid 232 . . 3 ((𝑎𝑃𝑏𝑃) → ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏))
2726rgen2a 3158 . 2 𝑎𝑃𝑏𝑃 ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)
28 dff13 6740 . 2 (𝐹:𝑃1-1𝑅 ↔ (𝐹:𝑃𝑅 ∧ ∀𝑎𝑃𝑏𝑃 ((𝐹𝑎) = (𝐹𝑏) → 𝑎 = 𝑏)))
294, 27, 28mpbir2an 703 1 𝐹:𝑃1-1𝑅
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  wrex 3090  {crab 3093  wss 3769  𝒫 cpw 4349  {cpr 4370   class class class wbr 4843  {copab 4905  cmpt 4922   × cxp 5310  wf 6097  1-1wf1 6098  cfv 6101  Pairscspr 42526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-spr 42527
This theorem is referenced by:  sprsymrelf1o  42547
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