Theorem 2nd1st 6352

Description: Swap the members of an ordered pair. (Contributed by NM, 31-Dec-2014.)

Assertion

Ref	Expression
2nd1st	|- ( A e. ( B X. C ) -> U. `' { A } = <. ( 2nd ` A ) , ( 1st ` A ) >. )

Proof of Theorem 2nd1st

Step	Hyp	Ref	Expression
1		1st2nd2 6347	. . . . 5 |- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2	1	sneqd 3686	. . . 4 |- ( A e. ( B X. C ) -> { A } = { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
3	2	cnveqd 4912	. . 3 |- ( A e. ( B X. C ) -> `' { A } = `' { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
4	3	unieqd 3909	. 2 |- ( A e. ( B X. C ) -> U. `' { A } = U. `' { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
5		1stexg 6339	. . 3 |- ( A e. ( B X. C ) -> ( 1st ` A ) e. _V )
6		2ndexg 6340	. . 3 |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. _V )
7		opswapg 5230	. . 3 |- ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) -> U. `' { <. ( 1st ` A ) , ( 2nd ` A ) >. } = <. ( 2nd ` A ) , ( 1st ` A ) >. )
8	5, 6, 7	syl2anc 411	. 2 |- ( A e. ( B X. C ) -> U. `' { <. ( 1st ` A ) , ( 2nd ` A ) >. } = <. ( 2nd ` A ) , ( 1st ` A ) >. )
9	4, 8	eqtrd 2264	1 |- ( A e. ( B X. C ) -> U. `' { A } = <. ( 2nd ` A ) , ( 1st ` A ) >. )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   {csn 3673   <.cop 3676   U.cuni 3898   X. cxp 4729   `'ccnv 4730   ` cfv 5333   1stc1st 6310   2ndc2nd 6311

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-1st 6312  df-2nd 6313

This theorem is referenced by: (None)