Theorem coepr 36067

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 3457	. . . . . 6 ⊢ 𝑥 ∈ V
3	1, 2	brcnv 5852	. . . . 5 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 E 𝐴)
4	1	epeli 5547	. . . . 5 ⊢ (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)
5	3, 4	bitri 277	. . . 4 ⊢ (𝐴◡ E 𝑥 ↔ 𝑥 ∈ 𝐴)
6	5	anbi1i 633	. . 3 ⊢ ((𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵))
7	6	exbii 1867	. 2 ⊢ (∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵))
8		coep.2	. . 3 ⊢ 𝐵 ∈ V
9	1, 8	brco 5840	. 2 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥(𝐴◡ E 𝑥 ∧ 𝑥𝑅𝐵))
10		df-rex 3086	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥𝑅𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥𝑅𝐵))
11	7, 9, 10	3bitr4i 305	1 ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)

Syntax hints:   ↔ wb 208   ∧ wa 399  ∃wex 1798   ∈ wcel 2141  ∃wrex 3085  Vcvv 3453   class class class wbr 5099   E cep 5544  ^◡ccnv 5644   ∘ ccom 5649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-eprel 5545  df-cnv 5653  df-co 5654

This theorem is referenced by:  elfuns  36227  brub  36268