Theorem brcnvepres 36029

Description: Restricted converse epsilon binary relation. (Contributed by Peter Mazsa, 10-Feb-2018.)

Assertion

Ref	Expression
brcnvepres	⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(◡ E ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))

Proof of Theorem brcnvepres

Step	Hyp	Ref	Expression
1		brres 5832	. 2 ⊢ (𝐶 ∈ 𝑊 → (𝐵(◡ E ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝐶)))
2		brcnvep 36027	. . 3 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡ E 𝐶 ↔ 𝐶 ∈ 𝐵))
3	2	anbi2d 632	. 2 ⊢ (𝐵 ∈ 𝑉 → ((𝐵 ∈ 𝐴 ∧ 𝐵◡ E 𝐶) ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))
4	1, 3	sylan9bbr 514	1 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊) → (𝐵(◡ E ↾ 𝐴)𝐶 ↔ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2114   class class class wbr 5030   E cep 5433  ^◡ccnv 5524   ↾ cres 5527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-rex 3059  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-eprel 5434  df-xp 5531  df-rel 5532  df-cnv 5533  df-res 5537

This theorem is referenced by:  dfeldisj3  36453  dfeldisj4  36454