Theorem dfcnv2 32607

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 6078	. 2 ⊢ Rel ◡𝑅
2		relxp 5659	. . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥}))
3	2	rgenw 3049	. . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))
4		reliun 5782	. . 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 3454	. . . . . . . . 9 ⊢ 𝑧 ∈ V
7		vex 3454	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	6, 7	opeldm 5874	. . . . . . . 8 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅)
9		df-rn 5652	. . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅
10	8, 9	eleqtrrdi 2840	. . . . . . 7 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅)
11		ssel2 3944	. . . . . . 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 6064	. . . . 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 4602	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
19	18	imaeq2d 6034	. . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧}))
20	19	opeliunxp2 5805	. . 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 5758	1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045   ⊆ wss 3917  {csn 4592  ⟨cop 4598  ∪ ciun 4958   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   “ cima 5644  Rel wrel 5646

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-sep 5254  ax-nul 5264  ax-pr 5390

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-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  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-sn 4593  df-pr 4595  df-op 4599  df-iun 4960  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654

This theorem is referenced by:  gsummpt2co  32995