Theorem 2nd1st 6279

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 6274	. . . . 5 |- ( A e. ( B X. C ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
2	1	sneqd 3651	. . . 4 |- ( A e. ( B X. C ) -> { A } = { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
3	2	cnveqd 4862	. . 3 |- ( A e. ( B X. C ) -> `' { A } = `' { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
4	3	unieqd 3867	. 2 |- ( A e. ( B X. C ) -> U. `' { A } = U. `' { <. ( 1st ` A ) , ( 2nd ` A ) >. } )
5		1stexg 6266	. . 3 |- ( A e. ( B X. C ) -> ( 1st ` A ) e. _V )
6		2ndexg 6267	. . 3 |- ( A e. ( B X. C ) -> ( 2nd ` A ) e. _V )
7		opswapg 5178	. . 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 2239	1 |- ( A e. ( B X. C ) -> U. `' { A } = <. ( 2nd ` A ) , ( 1st ` A ) >. )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   _Vcvv 2773   {csn 3638   <.cop 3641   U.cuni 3856   X. cxp 4681   `'ccnv 4682   ` cfv 5280   1stc1st 6237   2ndc2nd 6238

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3003  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-fo 5286  df-fv 5288  df-1st 6239  df-2nd 6240

This theorem is referenced by: (None)