Theorem mpt2sn 6887

Description: An operation (in maps-to notation) on two singletons. (Contributed by AV, 4-Aug-2019.)

Hypotheses

Ref	Expression
mpt2sn.f	|- F = ( x e. { A } , y e. { B } |-> C )
mpt2sn.a	|- ( x = A -> C = D )
mpt2sn.b	|- ( y = B -> D = E )

Assertion

Ref	Expression
mpt2sn	|- ( ( A e. V /\ B e. W /\ E e. U ) -> F = { <. <. A , B >. , E >. } )

Distinct variable groups:   x, A, y   x, B, y   x, E, y   x, U, y   x, V, y   x, W, y

Allowed substitution hints:   C( x, y)   D( x, y)   F( x, y)

Proof of Theorem mpt2sn

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpsng 6065	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( { A } X. { B } ) = { <. A , B >. } )
2	1	3adant3 1028	. . 3 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( { A } X. { B } ) = { <. A , B >. } )
3	2	mpteq1d 4484	. 2 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( p e. ( { A } X. { B } ) |-> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C ) = ( p e. { <. A , B >. } |-> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C ) )
4		mpt2sn.f	. . . 4 |- F = ( x e. { A } , y e. { B } |-> C )
5		mpt2mpts 6857	. . . 4 |- ( x e. { A } , y e. { B } |-> C ) = ( p e. ( { A } X. { B } ) |-> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C )
6	4, 5	eqtri 2473	. . 3 |- F = ( p e. ( { A } X. { B } ) |-> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C )
7	6	a1i 11	. 2 |- ( ( A e. V /\ B e. W /\ E e. U ) -> F = ( p e. ( { A } X. { B } ) |-> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C ) )
8		op2ndg 6806	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )
9		fveq2 5865	. . . . . . . . 9 |- ( p = <. A , B >. -> ( 2nd ` p ) = ( 2nd ` <. A , B >. ) )
10	9	eqcomd 2457	. . . . . . . 8 |- ( p = <. A , B >. -> ( 2nd ` <. A , B >. ) = ( 2nd ` p ) )
11	10	eqeq1d 2453	. . . . . . 7 |- ( p = <. A , B >. -> ( ( 2nd ` <. A , B >. ) = B <-> ( 2nd ` p ) = B ) )
12	8, 11	syl5ibcom 224	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( p = <. A , B >. -> ( 2nd ` p ) = B ) )
13	12	3adant3 1028	. . . . 5 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( p = <. A , B >. -> ( 2nd ` p ) = B ) )
14	13	imp 431	. . . 4 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> ( 2nd ` p ) = B )
15		op1stg 6805	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )
16		fveq2 5865	. . . . . . . . 9 |- ( p = <. A , B >. -> ( 1st ` p ) = ( 1st ` <. A , B >. ) )
17	16	eqcomd 2457	. . . . . . . 8 |- ( p = <. A , B >. -> ( 1st ` <. A , B >. ) = ( 1st ` p ) )
18	17	eqeq1d 2453	. . . . . . 7 |- ( p = <. A , B >. -> ( ( 1st ` <. A , B >. ) = A <-> ( 1st ` p ) = A ) )
19	15, 18	syl5ibcom 224	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( p = <. A , B >. -> ( 1st ` p ) = A ) )
20	19	3adant3 1028	. . . . 5 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( p = <. A , B >. -> ( 1st ` p ) = A ) )
21	20	imp 431	. . . 4 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> ( 1st ` p ) = A )
22		simp1 1008	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ E e. U ) -> A e. V )
23		simpl2 1012	. . . . . . . 8 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ x = A ) -> B e. W )
24		mpt2sn.a	. . . . . . . . . 10 |- ( x = A -> C = D )
25	24	adantl 468	. . . . . . . . 9 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ x = A ) -> C = D )
26		mpt2sn.b	. . . . . . . . 9 |- ( y = B -> D = E )
27	25, 26	sylan9eq 2505	. . . . . . . 8 |- ( ( ( ( A e. V /\ B e. W /\ E e. U ) /\ x = A ) /\ y = B ) -> C = E )
28	23, 27	csbied 3390	. . . . . . 7 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ x = A ) -> [_ B / y ]_ C = E )
29	22, 28	csbied 3390	. . . . . 6 |- ( ( A e. V /\ B e. W /\ E e. U ) -> [_ A / x ]_ [_ B / y ]_ C = E )
30	29	adantr 467	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> [_ A / x ]_ [_ B / y ]_ C = E )
31		csbeq1 3366	. . . . . . . 8 |- ( ( 1st ` p ) = A -> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C )
32	31	eqeq1d 2453	. . . . . . 7 |- ( ( 1st ` p ) = A -> ( [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E ) )
33	32	adantl 468	. . . . . 6 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> ( [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E ) )
34		csbeq1 3366	. . . . . . . . 9 |- ( ( 2nd ` p ) = B -> [_ ( 2nd ` p ) / y ]_ C = [_ B / y ]_ C )
35	34	adantr 467	. . . . . . . 8 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> [_ ( 2nd ` p ) / y ]_ C = [_ B / y ]_ C )
36	35	csbeq2dv 3781	. . . . . . 7 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = [_ A / x ]_ [_ B / y ]_ C )
37	36	eqeq1d 2453	. . . . . 6 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> ( [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ B / y ]_ C = E ) )
38	33, 37	bitrd 257	. . . . 5 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> ( [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ B / y ]_ C = E ) )
39	30, 38	syl5ibrcom 226	. . . 4 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E ) )
40	14, 21, 39	mp2and 685	. . 3 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E )
41		opex 4664	. . . 4 |- <. A , B >. e. _V
42	41	a1i 11	. . 3 |- ( ( A e. V /\ B e. W /\ E e. U ) -> <. A , B >. e. _V )
43		simp3 1010	. . 3 |- ( ( A e. V /\ B e. W /\ E e. U ) -> E e. U )
44	40, 42, 43	fmptsnd 6086	. 2 |- ( ( A e. V /\ B e. W /\ E e. U ) -> { <. <. A , B >. , E >. } = ( p e. { <. A , B >. } |-> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C ) )
45	3, 7, 44	3eqtr4d 2495	1 |- ( ( A e. V /\ B e. W /\ E e. U ) -> F = { <. <. A , B >. , E >. } )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   _Vcvv 3045   [_csb 3363   {csn 3968   <.cop 3974   |-> cmpt 4461   X. cxp 4832   ` cfv 5582   |-> cmpt2 6292   1stc1st 6791   2ndc2nd 6792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794

This theorem is referenced by:  mat1dim0  19498  mat1dimid  19499  mat1dimmul  19501  d1mat2pmat  19763