Theorem uncov 33390

Description: Value of uncurrying. (Contributed by Brendan Leahy, 2-Jun-2021.)

Assertion

Ref	Expression
uncov	|- ( ( A e. V /\ B e. W ) -> ( Auncurry F B ) = ( ( F ` A ) ` B ) )

Proof of Theorem uncov

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-br 4654	. . . . 5 |- ( <. A , B >.uncurry F w <-> <. <. A , B >. , w >. e. uncurry F )
2		df-unc 7394	. . . . . 6 |- uncurry F = { <. <. x , y >. , z >. | y ( F ` x ) z }
3	2	eleq2i 2693	. . . . 5 |- ( <. <. A , B >. , w >. e. uncurry F <-> <. <. A , B >. , w >. e. { <. <. x , y >. , z >. | y ( F ` x ) z } )
4	1, 3	bitri 264	. . . 4 |- ( <. A , B >.uncurry F w <-> <. <. A , B >. , w >. e. { <. <. x , y >. , z >. | y ( F ` x ) z } )
5		vex 3203	. . . . 5 |- w e. _V
6		simp2 1062	. . . . . . 7 |- ( ( x = A /\ y = B /\ z = w ) -> y = B )
7		fveq2 6191	. . . . . . . 8 |- ( x = A -> ( F ` x ) = ( F ` A ) )
8	7	3ad2ant1 1082	. . . . . . 7 |- ( ( x = A /\ y = B /\ z = w ) -> ( F ` x ) = ( F ` A ) )
9		simp3 1063	. . . . . . 7 |- ( ( x = A /\ y = B /\ z = w ) -> z = w )
10	6, 8, 9	breq123d 4667	. . . . . 6 |- ( ( x = A /\ y = B /\ z = w ) -> ( y ( F ` x ) z <-> B ( F ` A ) w ) )
11	10	eloprabga 6747	. . . . 5 |- ( ( A e. V /\ B e. W /\ w e. _V ) -> ( <. <. A , B >. , w >. e. { <. <. x , y >. , z >. | y ( F ` x ) z } <-> B ( F ` A ) w ) )
12	5, 11	mp3an3 1413	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( <. <. A , B >. , w >. e. { <. <. x , y >. , z >. | y ( F ` x ) z } <-> B ( F ` A ) w ) )
13	4, 12	syl5bb 272	. . 3 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >.uncurry F w <-> B ( F ` A ) w ) )
14	13	iotabidv 5872	. 2 |- ( ( A e. V /\ B e. W ) -> ( iota w <. A , B >.uncurry F w ) = ( iota w B ( F ` A ) w ) )
15		df-ov 6653	. . 3 |- ( Auncurry F B ) = (uncurry F ` <. A , B >. )
16		df-fv 5896	. . 3 |- (uncurry F ` <. A , B >. ) = ( iota w <. A , B >.uncurry F w )
17	15, 16	eqtri 2644	. 2 |- ( Auncurry F B ) = ( iota w <. A , B >.uncurry F w )
18		df-fv 5896	. 2 |- ( ( F ` A ) ` B ) = ( iota w B ( F ` A ) w )
19	14, 17, 18	3eqtr4g 2681	1 |- ( ( A e. V /\ B e. W ) -> ( Auncurry F B ) = ( ( F ` A ) ` B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   iotacio 5849   ` cfv 5888  (class class class)co 6650   {coprab 6651  uncurry cunc 7392

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  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-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-unc 7394

This theorem is referenced by:  curunc  33391  matunitlindflem2  33406