Theorem dfcnv2 31888

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 6100	. 2 ⊢ Rel ◡𝑅
2		relxp 5693	. . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥}))
3	2	rgenw 3065	. . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))
4		reliun 5814	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥})))
5	3, 4	mpbir 230	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))
6		vex 3478	. . . . . . . . 9 ⊢ 𝑧 ∈ V
7		vex 3478	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	6, 7	opeldm 5905	. . . . . . . 8 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅)
9		df-rn 5686	. . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅
10	8, 9	eleqtrrdi 2844	. . . . . . 7 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅)
11		ssel2 3976	. . . . . . 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 413	. . . . 5 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ 𝐴))
14	13	pm4.71rd 563	. . . 4 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅)))
15	6, 7	elimasn 6085	. . . . 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 288	. . 3 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧}))))
18		sneq 4637	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
19	18	imaeq2d 6057	. . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧}))
20	19	opeliunxp2 5836	. . 3 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})))
21	17, 20	bitr4di 288	. 2 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))))
22	1, 5, 21	eqrelrdv 5790	1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ⊆ wss 3947  {csn 4627  ⟨cop 4633  ∪ ciun 4996   × cxp 5673  ^◡ccnv 5674  dom cdm 5675  ran crn 5676   “ cima 5678  Rel wrel 5680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-iun 4998  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688

This theorem is referenced by:  gsummpt2co  32187