Theorem opfv 29448

Description: Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Assertion

Ref	Expression
opfv	|- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( F ` x ) = <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. )

Proof of Theorem opfv

Step	Hyp	Ref	Expression
1		simplr 792	. . . 4 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ran F C_ ( _V X. _V ) )
2		fvelrn 6352	. . . . 5 |- ( ( Fun F /\ x e. dom F ) -> ( F ` x ) e. ran F )
3	2	adantlr 751	. . . 4 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( F ` x ) e. ran F )
4	1, 3	sseldd 3604	. . 3 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( F ` x ) e. ( _V X. _V ) )
5		1st2ndb 7206	. . 3 |- ( ( F ` x ) e. ( _V X. _V ) <-> ( F ` x ) = <. ( 1st ` ( F ` x ) ) , ( 2nd ` ( F ` x ) ) >. )
6	4, 5	sylib 208	. 2 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( F ` x ) = <. ( 1st ` ( F ` x ) ) , ( 2nd ` ( F ` x ) ) >. )
7		fvco 6274	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> ( ( 1st o. F ) ` x ) = ( 1st ` ( F ` x ) ) )
8		fvco 6274	. . . 4 |- ( ( Fun F /\ x e. dom F ) -> ( ( 2nd o. F ) ` x ) = ( 2nd ` ( F ` x ) ) )
9	7, 8	opeq12d 4410	. . 3 |- ( ( Fun F /\ x e. dom F ) -> <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. = <. ( 1st ` ( F ` x ) ) , ( 2nd ` ( F ` x ) ) >. )
10	9	adantlr 751	. 2 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. = <. ( 1st ` ( F ` x ) ) , ( 2nd ` ( F ` x ) ) >. )
11	6, 10	eqtr4d 2659	1 |- ( ( ( Fun F /\ ran F C_ ( _V X. _V ) ) /\ x e. dom F ) -> ( F ` x ) = <. ( ( 1st o. F ) ` x ) , ( ( 2nd o. F ) ` x ) >. )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cop 4183   X. cxp 5112   dom cdm 5114   ran crn 5115   o. ccom 5118   Fun wfun 5882   ` cfv 5888   1stc1st 7166   2ndc2nd 7167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  xppreima  29449  xppreima2  29450