Theorem brswap 5510

Description: Binary relationship of Swap . (Contributed by SF, 23-Feb-2015.)

Assertion

Ref	Expression
brswap	⊢ (A Swap B ↔ ∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨y, x⟩))

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

Proof of Theorem brswap

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. 2 ⊢ (A Swap B → (A ∈ V ∧ B ∈ V))
2		vex 2863	. . . . . 6 ⊢ x ∈ V
3		vex 2863	. . . . . 6 ⊢ y ∈ V
4	2, 3	opex 4589	. . . . 5 ⊢ ⟨x, y⟩ ∈ V
5		eleq1 2413	. . . . 5 ⊢ (A = ⟨x, y⟩ → (A ∈ V ↔ ⟨x, y⟩ ∈ V))
6	4, 5	mpbiri 224	. . . 4 ⊢ (A = ⟨x, y⟩ → A ∈ V)
7	3, 2	opex 4589	. . . . 5 ⊢ ⟨y, x⟩ ∈ V
8		eleq1 2413	. . . . 5 ⊢ (B = ⟨y, x⟩ → (B ∈ V ↔ ⟨y, x⟩ ∈ V))
9	7, 8	mpbiri 224	. . . 4 ⊢ (B = ⟨y, x⟩ → B ∈ V)
10	6, 9	anim12i 549	. . 3 ⊢ ((A = ⟨x, y⟩ ∧ B = ⟨y, x⟩) → (A ∈ V ∧ B ∈ V))
11	10	exlimivv 1635	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨y, x⟩) → (A ∈ V ∧ B ∈ V))
12		eqeq1 2359	. . . . 5 ⊢ (a = A → (a = ⟨x, y⟩ ↔ A = ⟨x, y⟩))
13	12	anbi1d 685	. . . 4 ⊢ (a = A → ((a = ⟨x, y⟩ ∧ b = ⟨y, x⟩) ↔ (A = ⟨x, y⟩ ∧ b = ⟨y, x⟩)))
14	13	2exbidv 1628	. . 3 ⊢ (a = A → (∃x∃y(a = ⟨x, y⟩ ∧ b = ⟨y, x⟩) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ b = ⟨y, x⟩)))
15		eqeq1 2359	. . . . 5 ⊢ (b = B → (b = ⟨y, x⟩ ↔ B = ⟨y, x⟩))
16	15	anbi2d 684	. . . 4 ⊢ (b = B → ((A = ⟨x, y⟩ ∧ b = ⟨y, x⟩) ↔ (A = ⟨x, y⟩ ∧ B = ⟨y, x⟩)))
17	16	2exbidv 1628	. . 3 ⊢ (b = B → (∃x∃y(A = ⟨x, y⟩ ∧ b = ⟨y, x⟩) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨y, x⟩)))
18		df-swap 4725	. . 3 ⊢ Swap = {⟨a, b⟩ ∣ ∃x∃y(a = ⟨x, y⟩ ∧ b = ⟨y, x⟩)}
19	14, 17, 18	brabg 4707	. 2 ⊢ ((A ∈ V ∧ B ∈ V) → (A Swap B ↔ ∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨y, x⟩)))
20	1, 11, 19	pm5.21nii 342	1 ⊢ (A Swap B ↔ ∃x∃y(A = ⟨x, y⟩ ∧ B = ⟨y, x⟩))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ⟨cop 4562   class class class wbr 4640   Swap cswap 4719

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-swap 4725

This theorem is referenced by:  cnvswap  5511