Theorem mpt2sn 7268

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 6406	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( { A } X. { B } ) = { <. A , B >. } )
2	1	3adant3 1081	. . 3 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( { A } X. { B } ) = { <. A , B >. } )
3	2	mpteq1d 4738	. 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 7234	. . . 4 |- ( x e. { A } , y e. { B } |-> C ) = ( p e. ( { A } X. { B } ) |-> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C )
6	4, 5	eqtri 2644	. . 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 7181	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )
9		fveq2 6191	. . . . . . . . 9 |- ( p = <. A , B >. -> ( 2nd ` p ) = ( 2nd ` <. A , B >. ) )
10	9	eqcomd 2628	. . . . . . . 8 |- ( p = <. A , B >. -> ( 2nd ` <. A , B >. ) = ( 2nd ` p ) )
11	10	eqeq1d 2624	. . . . . . 7 |- ( p = <. A , B >. -> ( ( 2nd ` <. A , B >. ) = B <-> ( 2nd ` p ) = B ) )
12	8, 11	syl5ibcom 235	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( p = <. A , B >. -> ( 2nd ` p ) = B ) )
13	12	3adant3 1081	. . . . 5 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( p = <. A , B >. -> ( 2nd ` p ) = B ) )
14	13	imp 445	. . . 4 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> ( 2nd ` p ) = B )
15		op1stg 7180	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )
16		fveq2 6191	. . . . . . . . 9 |- ( p = <. A , B >. -> ( 1st ` p ) = ( 1st ` <. A , B >. ) )
17	16	eqcomd 2628	. . . . . . . 8 |- ( p = <. A , B >. -> ( 1st ` <. A , B >. ) = ( 1st ` p ) )
18	17	eqeq1d 2624	. . . . . . 7 |- ( p = <. A , B >. -> ( ( 1st ` <. A , B >. ) = A <-> ( 1st ` p ) = A ) )
19	15, 18	syl5ibcom 235	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( p = <. A , B >. -> ( 1st ` p ) = A ) )
20	19	3adant3 1081	. . . . 5 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( p = <. A , B >. -> ( 1st ` p ) = A ) )
21	20	imp 445	. . . 4 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> ( 1st ` p ) = A )
22		simp1 1061	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ E e. U ) -> A e. V )
23		simpl2 1065	. . . . . . . 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 482	. . . . . . . . 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 2676	. . . . . . . 8 |- ( ( ( ( A e. V /\ B e. W /\ E e. U ) /\ x = A ) /\ y = B ) -> C = E )
28	23, 27	csbied 3560	. . . . . . 7 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ x = A ) -> [_ B / y ]_ C = E )
29	22, 28	csbied 3560	. . . . . 6 |- ( ( A e. V /\ B e. W /\ E e. U ) -> [_ A / x ]_ [_ B / y ]_ C = E )
30	29	adantr 481	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> [_ A / x ]_ [_ B / y ]_ C = E )
31		csbeq1 3536	. . . . . . . 8 |- ( ( 1st ` p ) = A -> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C )
32	31	eqeq1d 2624	. . . . . . 7 |- ( ( 1st ` p ) = A -> ( [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E ) )
33	32	adantl 482	. . . . . 6 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> ( [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E ) )
34		csbeq1 3536	. . . . . . . . 9 |- ( ( 2nd ` p ) = B -> [_ ( 2nd ` p ) / y ]_ C = [_ B / y ]_ C )
35	34	adantr 481	. . . . . . . 8 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> [_ ( 2nd ` p ) / y ]_ C = [_ B / y ]_ C )
36	35	csbeq2dv 3992	. . . . . . 7 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = [_ A / x ]_ [_ B / y ]_ C )
37	36	eqeq1d 2624	. . . . . 6 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> ( [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ B / y ]_ C = E ) )
38	33, 37	bitrd 268	. . . . 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 237	. . . 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 715	. . 3 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E )
41		opex 4932	. . . 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 1063	. . 3 |- ( ( A e. V /\ B e. W /\ E e. U ) -> E e. U )
44	40, 42, 43	fmptsnd 6435	. 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 2666	1 |- ( ( A e. V /\ B e. W /\ E e. U ) -> F = { <. <. A , B >. , E >. } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   [_csb 3533   {csn 4177   <.cop 4183   |-> cmpt 4729   X. cxp 5112   ` cfv 5888   |-> cmpt2 6652   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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169

This theorem is referenced by:  mat1dim0  20279  mat1dimid  20280  mat1dimmul  20282  d1mat2pmat  20544