Theorem brswap 5510

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

Assertion

Ref	Expression
brswap	|- (A Swap B <-> E.xE.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 e. _V /\ B e. _V))
2		vex 2863	. . . . . 6 |- x e. _V
3		vex 2863	. . . . . 6 |- y e. _V
4	2, 3	opex 4589	. . . . 5 |- <.x, y>. e. _V
5		eleq1 2413	. . . . 5 |- (A = <.x, y>. -> (A e. _V <-> <.x, y>. e. _V))
6	4, 5	mpbiri 224	. . . 4 |- (A = <.x, y>. -> A e. _V)
7	3, 2	opex 4589	. . . . 5 |- <.y, x>. e. _V
8		eleq1 2413	. . . . 5 |- (B = <.y, x>. -> (B e. _V <-> <.y, x>. e. _V))
9	7, 8	mpbiri 224	. . . 4 |- (B = <.y, x>. -> B e. _V)
10	6, 9	anim12i 549	. . 3 |- ((A = <.x, y>. /\ B = <.y, x>.) -> (A e. _V /\ B e. _V))
11	10	exlimivv 1635	. 2 |- (E.xE.y(A = <.x, y>. /\ B = <.y, x>.) -> (A e. _V /\ B e. _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 -> (E.xE.y(a = <.x, y>. /\ b = <.y, x>.) <-> E.xE.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 -> (E.xE.y(A = <.x, y>. /\ b = <.y, x>.) <-> E.xE.y(A = <.x, y>. /\ B = <.y, x>.)))
18		df-swap 4725	. . 3 |- Swap = {<.a, b>. | E.xE.y(a = <.x, y>. /\ b = <.y, x>.)}
19	14, 17, 18	brabg 4707	. 2 |- ((A e. _V /\ B e. _V) -> (A Swap B <-> E.xE.y(A = <.x, y>. /\ B = <.y, x>.)))
20	1, 11, 19	pm5.21nii 342	1 |- (A Swap B <-> E.xE.y(A = <.x, y>. /\ B = <.y, x>.))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. 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