Theorem sprsymrelfv 47486

Description: The value of the function 𝐹 which maps a subset of the set of pairs over a fixed set 𝑉 to the relation relating two elements of the set 𝑉 iff they are in a pair of the subset. (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
sprsymrelfv	⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}})

Distinct variable groups:   𝑃,𝑝   𝑉,𝑐,𝑥,𝑦   𝑝,𝑐,𝑥,𝑦,𝑋

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

Proof of Theorem sprsymrelfv

Step	Hyp	Ref	Expression
1		sprsymrelf.f	. 2 ⊢ 𝐹 = (𝑝 ∈ 𝑃 ↦ {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}})
2		rexeq 3321	. . 3 ⊢ (𝑝 = 𝑋 → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}))
3	2	opabbidv 5208	. 2 ⊢ (𝑝 = 𝑋 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦}} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}})
4		id 22	. 2 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ∈ 𝑃)
5		elpwi 4606	. . . 4 ⊢ (𝑋 ∈ 𝒫 (Pairs‘𝑉) → 𝑋 ⊆ (Pairs‘𝑉))
6		sprsymrelf.p	. . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
7	5, 6	eleq2s 2858	. . 3 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ⊆ (Pairs‘𝑉))
8		sprsymrelfvlem 47482	. . 3 ⊢ (𝑋 ⊆ (Pairs‘𝑉) → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
9	7, 8	syl 17	. 2 ⊢ (𝑋 ∈ 𝑃 → {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}} ∈ 𝒫 (𝑉 × 𝑉))
10	1, 3, 4, 9	fvmptd3 7038	1 ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  {crab 3435   ⊆ wss 3950  𝒫 cpw 4599  {cpr 4627   class class class wbr 5142  {copab 5204   ↦ cmpt 5224   × cxp 5682  ‘cfv 6560  Pairscspr 47469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-spr 47470

This theorem is referenced by:  sprsymrelf1  47488  sprsymrelfo  47489