Theorem dfcnv2 32686

Description: Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)

Assertion

Ref	Expression
dfcnv2	⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfcnv2

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 6122	. 2 ⊢ Rel ◡𝑅
2		relxp 5703	. . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥}))
3	2	rgenw 3065	. . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))
4		reliun 5826	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥})))
5	3, 4	mpbir 231	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))
6		vex 3484	. . . . . . . . 9 ⊢ 𝑧 ∈ V
7		vex 3484	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	6, 7	opeldm 5918	. . . . . . . 8 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅)
9		df-rn 5696	. . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅
10	8, 9	eleqtrrdi 2852	. . . . . . 7 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅)
11		ssel2 3978	. . . . . . 7 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 𝑧 ∈ ran 𝑅) → 𝑧 ∈ 𝐴)
12	10, 11	sylan2 593	. . . . . 6 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅) → 𝑧 ∈ 𝐴)
13	12	ex 412	. . . . 5 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ 𝐴))
14	13	pm4.71rd 562	. . . 4 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅)))
15	6, 7	elimasn 6108	. . . . 5 ⊢ (𝑦 ∈ (◡𝑅 “ {𝑧}) ↔ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅)
16	15	anbi2i 623	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})) ↔ (𝑧 ∈ 𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅))
17	14, 16	bitr4di 289	. . 3 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧}))))
18		sneq 4636	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
19	18	imaeq2d 6078	. . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧}))
20	19	opeliunxp2 5849	. . 3 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})))
21	17, 20	bitr4di 289	. 2 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))))
22	1, 5, 21	eqrelrdv 5802	1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   ⊆ wss 3951  {csn 4626  ⟨cop 4632  ∪ ciun 4991   × cxp 5683  ^◡ccnv 5684  dom cdm 5685  ran crn 5686   “ cima 5688  Rel wrel 5690

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-iun 4993  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698

This theorem is referenced by:  gsummpt2co  33051