Theorem cnvdif 5035

Description: Distributive law for converse over set difference. (Contributed by set.mm contributors, 26-Jun-2014.)

Assertion

Ref	Expression
cnvdif	⊢ ◡(A ∖ B) = (◡A ∖ ◡B)

Proof of Theorem cnvdif

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldif 3222	. . . 4 ⊢ (⟨y, x⟩ ∈ (A ∖ B) ↔ (⟨y, x⟩ ∈ A ∧ ¬ ⟨y, x⟩ ∈ B))
2		opelcnv 4894	. . . . 5 ⊢ (⟨x, y⟩ ∈ ◡A ↔ ⟨y, x⟩ ∈ A)
3		opelcnv 4894	. . . . . 6 ⊢ (⟨x, y⟩ ∈ ◡B ↔ ⟨y, x⟩ ∈ B)
4	3	notbii 287	. . . . 5 ⊢ (¬ ⟨x, y⟩ ∈ ◡B ↔ ¬ ⟨y, x⟩ ∈ B)
5	2, 4	anbi12i 678	. . . 4 ⊢ ((⟨x, y⟩ ∈ ◡A ∧ ¬ ⟨x, y⟩ ∈ ◡B) ↔ (⟨y, x⟩ ∈ A ∧ ¬ ⟨y, x⟩ ∈ B))
6	1, 5	bitr4i 243	. . 3 ⊢ (⟨y, x⟩ ∈ (A ∖ B) ↔ (⟨x, y⟩ ∈ ◡A ∧ ¬ ⟨x, y⟩ ∈ ◡B))
7		opelcnv 4894	. . 3 ⊢ (⟨x, y⟩ ∈ ◡(A ∖ B) ↔ ⟨y, x⟩ ∈ (A ∖ B))
8		eldif 3222	. . 3 ⊢ (⟨x, y⟩ ∈ (◡A ∖ ◡B) ↔ (⟨x, y⟩ ∈ ◡A ∧ ¬ ⟨x, y⟩ ∈ ◡B))
9	6, 7, 8	3bitr4i 268	. 2 ⊢ (⟨x, y⟩ ∈ ◡(A ∖ B) ↔ ⟨x, y⟩ ∈ (◡A ∖ ◡B))
10	9	eqrelriv 4851	1 ⊢ ◡(A ∖ B) = (◡A ∖ ◡B)

Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207  ⟨cop 4562  ^◡ccnv 4772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-cnv 4786

This theorem is referenced by:  cnvin  5036