Theorem dfcnv2 32771

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 6073	. 2 ⊢ Rel ◡𝑅
2		relxp 5652	. . . 4 ⊢ Rel ({𝑥} × (◡𝑅 “ {𝑥}))
3	2	rgenw 3056	. . 3 ⊢ ∀𝑥 ∈ 𝐴 Rel ({𝑥} × (◡𝑅 “ {𝑥}))
4		reliun 5775	. . 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 3446	. . . . . . . . 9 ⊢ 𝑧 ∈ V
7		vex 3446	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	6, 7	opeldm 5866	. . . . . . . 8 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ dom ◡𝑅)
9		df-rn 5645	. . . . . . . 8 ⊢ ran 𝑅 = dom ◡𝑅
10	8, 9	eleqtrrdi 2848	. . . . . . 7 ⊢ (⟨𝑧, 𝑦⟩ ∈ ◡𝑅 → 𝑧 ∈ ran 𝑅)
11		ssel2 3930	. . . . . . 7 ⊢ ((ran 𝑅 ⊆ 𝐴 ∧ 𝑧 ∈ ran 𝑅) → 𝑧 ∈ 𝐴)
12	10, 11	sylan2 594	. . . . . 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 6059	. . . . 5 ⊢ (𝑦 ∈ (◡𝑅 “ {𝑧}) ↔ ⟨𝑧, 𝑦⟩ ∈ ◡𝑅)
16	15	anbi2i 624	. . . 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 4592	. . . . 5 ⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧})
19	18	imaeq2d 6029	. . . 4 ⊢ (𝑥 = 𝑧 → (◡𝑅 “ {𝑥}) = (◡𝑅 “ {𝑧}))
20	19	opeliunxp2 5797	. . 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 5751	1 ⊢ (ran 𝑅 ⊆ 𝐴 → ◡𝑅 = ∪ 𝑥 ∈ 𝐴 ({𝑥} × (◡𝑅 “ {𝑥})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3903  {csn 4582  ⟨cop 4588  ∪ ciun 4948   × cxp 5632  ^◡ccnv 5633  dom cdm 5634  ran crn 5635   “ cima 5637  Rel wrel 5639

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-iun 4950  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647

This theorem is referenced by:  gsummpt2co  33148