Theorem coepr 36223

Description: Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)

Hypotheses

Ref	Expression
coep.1	⊢ 𝐴 ∈ V
coep.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
coepr	⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅

Proof of Theorem coepr

Step	Hyp	Ref	Expression
1		coep.1	. . . . . 6 ⊢ 𝐴 ∈ V
2		vex 3459	. . . . . 6 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5870	. . . . 5 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 E 𝐴)
4	1	epeli 5565	. . . . 5 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
5	3, 4	bitri 278	. . . 4 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 ∈ 𝐴)
6	5	anbi1i 635	. . 3 ⊢ ((𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵))
7	6	exbii 1878	. 2 ⊢ (∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵))
8		coep.2	. . 3 ⊢ 𝐵 ∈ V
9	1, 8	brco 5858	. 2 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵))
10		df-rex 3090	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵))
11	7, 9, 10	3bitr4i 306	1 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1809   ∈ wcel 2143  ∃wrex 3089  Vcvv 3455   class class class wbr 5110   E cep 5562  ^◡ccnv 5662   ∘ ccom 5667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-br 5111  df-opab 5175  df-eprel 5563  df-cnv 5671  df-co 5672

This theorem is referenced by:  elfuns  36383  brub  36424