Theorem sprsymrelfv 47499

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 3297	. . 3 ⊢ (𝑝 = 𝑋 → (∃𝑐 ∈ 𝑝 𝑐 = {𝑥, 𝑦} ↔ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}))
3	2	opabbidv 5176	. 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 4573	. . . 4 ⊢ (𝑋 ∈ 𝒫 (Pairs‘𝑉) → 𝑋 ⊆ (Pairs‘𝑉))
6		sprsymrelf.p	. . . 4 ⊢ 𝑃 = 𝒫 (Pairs‘𝑉)
7	5, 6	eleq2s 2847	. . 3 ⊢ (𝑋 ∈ 𝑃 → 𝑋 ⊆ (Pairs‘𝑉))
8		sprsymrelfvlem 47495	. . 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 6994	1 ⊢ (𝑋 ∈ 𝑃 → (𝐹‘𝑋) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑐 ∈ 𝑋 𝑐 = {𝑥, 𝑦}})

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  {crab 3408   ⊆ wss 3917  𝒫 cpw 4566  {cpr 4594   class class class wbr 5110  {copab 5172   ↦ cmpt 5191   × cxp 5639  ‘cfv 6514  Pairscspr 47482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-spr 47483

This theorem is referenced by:  sprsymrelf1  47501  sprsymrelfo  47502