Theorem mpt2sn 6890

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 6073	. . . 4 |- ( ( A e. V /\ B e. W ) -> ( { A } X. { B } ) = { <. A , B >. } )
2	1	3adant3 1016	. . 3 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( { A } X. { B } ) = { <. A , B >. } )
3	2	mpteq1d 4538	. 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 6863	. . . 4 |- ( x e. { A } , y e. { B } |-> C ) = ( p e. ( { A } X. { B } ) |-> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C )
6	4, 5	eqtri 2486	. . 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 6812	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( 2nd ` <. A , B >. ) = B )
9		fveq2 5872	. . . . . . . . 9 |- ( p = <. A , B >. -> ( 2nd ` p ) = ( 2nd ` <. A , B >. ) )
10	9	eqcomd 2465	. . . . . . . 8 |- ( p = <. A , B >. -> ( 2nd ` <. A , B >. ) = ( 2nd ` p ) )
11	10	eqeq1d 2459	. . . . . . 7 |- ( p = <. A , B >. -> ( ( 2nd ` <. A , B >. ) = B <-> ( 2nd ` p ) = B ) )
12	8, 11	syl5ibcom 220	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( p = <. A , B >. -> ( 2nd ` p ) = B ) )
13	12	3adant3 1016	. . . . 5 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( p = <. A , B >. -> ( 2nd ` p ) = B ) )
14	13	imp 429	. . . 4 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> ( 2nd ` p ) = B )
15		op1stg 6811	. . . . . . 7 |- ( ( A e. V /\ B e. W ) -> ( 1st ` <. A , B >. ) = A )
16		fveq2 5872	. . . . . . . . 9 |- ( p = <. A , B >. -> ( 1st ` p ) = ( 1st ` <. A , B >. ) )
17	16	eqcomd 2465	. . . . . . . 8 |- ( p = <. A , B >. -> ( 1st ` <. A , B >. ) = ( 1st ` p ) )
18	17	eqeq1d 2459	. . . . . . 7 |- ( p = <. A , B >. -> ( ( 1st ` <. A , B >. ) = A <-> ( 1st ` p ) = A ) )
19	15, 18	syl5ibcom 220	. . . . . 6 |- ( ( A e. V /\ B e. W ) -> ( p = <. A , B >. -> ( 1st ` p ) = A ) )
20	19	3adant3 1016	. . . . 5 |- ( ( A e. V /\ B e. W /\ E e. U ) -> ( p = <. A , B >. -> ( 1st ` p ) = A ) )
21	20	imp 429	. . . 4 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> ( 1st ` p ) = A )
22		simp1 996	. . . . . . 7 |- ( ( A e. V /\ B e. W /\ E e. U ) -> A e. V )
23		simpl2 1000	. . . . . . . 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 466	. . . . . . . . 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 2518	. . . . . . . 8 |- ( ( ( ( A e. V /\ B e. W /\ E e. U ) /\ x = A ) /\ y = B ) -> C = E )
28	23, 27	csbied 3457	. . . . . . 7 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ x = A ) -> [_ B / y ]_ C = E )
29	22, 28	csbied 3457	. . . . . 6 |- ( ( A e. V /\ B e. W /\ E e. U ) -> [_ A / x ]_ [_ B / y ]_ C = E )
30	29	adantr 465	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> [_ A / x ]_ [_ B / y ]_ C = E )
31		csbeq1 3433	. . . . . . . 8 |- ( ( 1st ` p ) = A -> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C )
32	31	eqeq1d 2459	. . . . . . 7 |- ( ( 1st ` p ) = A -> ( [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E ) )
33	32	adantl 466	. . . . . 6 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> ( [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E ) )
34		csbeq1 3433	. . . . . . . . 9 |- ( ( 2nd ` p ) = B -> [_ ( 2nd ` p ) / y ]_ C = [_ B / y ]_ C )
35	34	adantr 465	. . . . . . . 8 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> [_ ( 2nd ` p ) / y ]_ C = [_ B / y ]_ C )
36	35	csbeq2dv 3843	. . . . . . 7 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = [_ A / x ]_ [_ B / y ]_ C )
37	36	eqeq1d 2459	. . . . . 6 |- ( ( ( 2nd ` p ) = B /\ ( 1st ` p ) = A ) -> ( [_ A / x ]_ [_ ( 2nd ` p ) / y ]_ C = E <-> [_ A / x ]_ [_ B / y ]_ C = E ) )
38	33, 37	bitrd 253	. . . . 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 222	. . . 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 679	. . 3 |- ( ( ( A e. V /\ B e. W /\ E e. U ) /\ p = <. A , B >. ) -> [_ ( 1st ` p ) / x ]_ [_ ( 2nd ` p ) / y ]_ C = E )
41		opex 4720	. . . 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 998	. . 3 |- ( ( A e. V /\ B e. W /\ E e. U ) -> E e. U )
44	40, 42, 43	fmptsnd 6094	. 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 2508	1 |- ( ( A e. V /\ B e. W /\ E e. U ) -> F = { <. <. A , B >. , E >. } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   _Vcvv 3109   [_csb 3430   {csn 4032   <.cop 4038   |-> cmpt 4515   X. cxp 5006   ` cfv 5594   |-> cmpt2 6298   1stc1st 6797   2ndc2nd 6798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800

This theorem is referenced by:  mat1dim0  19193  mat1dimid  19194  mat1dimmul  19196  d1mat2pmat  19458