Theorem eqopi 6151

Description: Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)

Assertion

Ref	Expression
eqopi	|- ( ( A e. ( V X. W ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) -> A = <. B , C >. )

Proof of Theorem eqopi

Step	Hyp	Ref	Expression
1		xpss 4719	. . 3 |- ( V X. W ) C_ ( _V X. _V )
2	1	sseli 3143	. 2 |- ( A e. ( V X. W ) -> A e. ( _V X. _V ) )
3		elxp6 6148	. . . 4 |- ( A e. ( _V X. _V ) <-> ( A = <. ( 1st ` A ) , ( 2nd ` A ) >. /\ ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) ) )
4	3	simplbi 272	. . 3 |- ( A e. ( _V X. _V ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
5		opeq12 3767	. . 3 |- ( ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) -> <. ( 1st ` A ) , ( 2nd ` A ) >. = <. B , C >. )
6	4, 5	sylan9eq 2223	. 2 |- ( ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) -> A = <. B , C >. )
7	2, 6	sylan 281	1 |- ( ( A e. ( V X. W ) /\ ( ( 1st ` A ) = B /\ ( 2nd ` A ) = C ) ) -> A = <. B , C >. )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   <.cop 3586   X. cxp 4609   ` cfv 5198   1stc1st 6117   2ndc2nd 6118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fv 5206  df-1st 6119  df-2nd 6120

This theorem is referenced by:  op1steq  6158  dfoprab3  6170  1stconst  6200  2ndconst  6201  cnvoprab  6213  upxp  13066