Theorem elcnv2 2515

Description: Membership in a converse. Equation (5) of [Suppes] p. 62.

Assertion

Ref	Expression
elcnv2	⊢ (A ∈ ◡R ↔ ∃x∃y(A = ⟨x, y⟩ ∧ ⟨y, x⟩ ∈ R))

Distinct variable group(s):   x,y,A   x,R,y

Proof of Theorem elcnv2

Step	Hyp	Ref	Expression
1		elcnv 2514	. 2 ⊢ (A ∈ ◡R ↔ ∃x∃y(A = ⟨x, y⟩ ∧ yRx))
2		df-br 2063	. . . 4 ⊢ (yRx ↔ ⟨y, x⟩ ∈ R)
3	2	anbi2i 367	. . 3 ⊢ ((A = ⟨x, y⟩ ∧ yRx) ↔ (A = ⟨x, y⟩ ∧ ⟨y, x⟩ ∈ R))
4	3	bi2ex 734	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ yRx) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ ⟨y, x⟩ ∈ R))
5	1, 4	bitr 151	1 ⊢ (A ∈ ◡R ↔ ∃x∃y(A = ⟨x, y⟩ ∧ ⟨y, x⟩ ∈ R))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   class class class wbr 2054  ^◡ccnv 2409

This theorem is referenced by:  cnvuni 2521

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426

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