Theorem cnvopab 4726

Description: The converse of a class abstraction of ordered pairs. (Contributed by NM, 11-Dec-2003.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cnvopab	|- `' { <. x , y >. | ph } = { <. y , x >. | ph }

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem cnvopab

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 4703	. 2 |- Rel `' { <. x , y >. | ph }
2		relopab 4464	. 2 |- Rel { <. y , x >. | ph }
3		opelopabsbALT 3996	. . . 4 |- ( <. w , z >. e. { <. x , y >. | ph } <-> [ z / y ] [ w / x ] ph )
4		sbcom2 1863	. . . 4 |- ( [ z / y ] [ w / x ] ph <-> [ w / x ] [ z / y ] ph )
5	3, 4	bitri 173	. . 3 |- ( <. w , z >. e. { <. x , y >. | ph } <-> [ w / x ] [ z / y ] ph )
6		vex 2560	. . . 4 |- z e. _V
7		vex 2560	. . . 4 |- w e. _V
8	6, 7	opelcnv 4517	. . 3 |- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. w , z >. e. { <. x , y >. | ph } )
9		opelopabsbALT 3996	. . 3 |- ( <. z , w >. e. { <. y , x >. | ph } <-> [ w / x ] [ z / y ] ph )
10	5, 8, 9	3bitr4i 201	. 2 |- ( <. z , w >. e. `' { <. x , y >. | ph } <-> <. z , w >. e. { <. y , x >. | ph } )
11	1, 2, 10	eqrelriiv 4434	1 |- `' { <. x , y >. | ph } = { <. y , x >. | ph }

Syntax hints:   = wceq 1243   e. wcel 1393   [wsb 1645   <.cop 3378   {copab 3817   `'ccnv 4344

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353

This theorem is referenced by:  cnvxp  4742  mptpreima  4814  f1ocnvd  5702