Theorem dfcnv2 33095

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 6108	. 2 ⊢ Rel ◡𝑅
2		relxp 5681	. . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥}))
3	2	rgenw 3085	. . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))
4		reliun 5805	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥})))
5	3, 4	mpbir 234	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))
6		vex 3461	. . . . . . . . 9 ⊢ 𝑧 ∈ V
7		vex 3461	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	6, 7	opeldm 5899	. . . . . . . 8 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅)
9		df-rn 5674	. . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅
10	8, 9	eleqtrrdi 2876	. . . . . . 7 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅)
11		ssel2 3933	. . . . . . 7 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 𝑧 ∈ ran 𝑅) → 𝑧 ∈ 𝐴)
12	10, 11	sylan2 605	. . . . . 6 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅) → 𝑧 ∈ 𝐴)
13	12	ex 418	. . . . 5 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ 𝐴))
14	13	pm4.71rd 572	. . . 4 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅)))
15	6, 7	elimasn 6094	. . . . 5 ⊢ (𝑦 ∈ (◡𝑅 “ {𝑧}) ↔ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅)
16	15	anbi2i 635	. . . 4 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})) ↔ (𝑧 ∈ 𝐴 ∧ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅))
17	14, 16	bitr4di 292	. . 3 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧}))))
18		sneq 4601	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
19	18	imaeq2d 6064	. . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧}))
20	19	opeliunxp2 5826	. . 3 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})) ↔ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ (◡𝑅 “ {𝑧})))
21	17, 20	bitr4di 292	. 2 ⊢ (ran 𝑅 ⊆ 𝐴 → (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 ↔ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥}))))
22	1, 5, 21	eqrelrdv 5780	1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081   ⊆ wss 3906  {csn 4591  ⟨cop 4597  ∪ ciun 4958   × cxp 5661  ^◡ccnv 5662  dom cdm 5663  ran crn 5664   “ cima 5666  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-iun 4960  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676

This theorem is used by:  gsummpt2co  33436