Theorem 2nd1st 6324

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 6319	. . . . 5 |- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2	1	sneqd 3679	. . . 4 |- ( A e. ( B X. C ) -> { A } = { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
3	2	cnveqd 4897	. . 3 |- ( A e. ( B X. C ) -> `' { A } = `' { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
4	3	unieqd 3898	. 2 |- ( A e. ( B X. C ) -> U. `' { A } = U. `' { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
5		1stexg 6311	. . 3 |- ( A e. ( B X. C ) -> ( 1st ` A ) e. _V )
6		2ndexg 6312	. . 3 |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. _V )
7		opswapg 5214	. . 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 2262	1 |- ( A e. ( B X. C ) -> U. `' { A } = <. ( 2nd ` A ) , ( 1st ` A ) >. )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   _Vcvv 2799   {csn 3666   <.cop 3669   U.cuni 3887   X. cxp 4716   `'ccnv 4717   ` cfv 5317   1stc1st 6282   2ndc2nd 6283

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fo 5323  df-fv 5325  df-1st 6284  df-2nd 6285

This theorem is referenced by: (None)