Theorem fmptsng 6434

Description: Express a singleton function in maps-to notation. Version of fmptsn 6433 allowing the mapping value to depend on the mapping variable (usual case). (Contributed by AV, 27-Feb-2019.)

Hypothesis

Ref	Expression
fmptsng.1	|- ( x = A -> B = C )

Assertion

Ref	Expression
fmptsng	|- ( ( A e. V /\ C e. W ) -> { <. A , C >. } = ( x e. { A } |-> B ) )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   B( x)   V( x)   W( x)

Proof of Theorem fmptsng

Dummy variables p y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 4193	. . . . 5 |- ( x e. { A } <-> x = A )
2	1	bicomi 214	. . . 4 |- ( x = A <-> x e. { A } )
3	2	anbi1i 731	. . 3 |- ( ( x = A /\ y = B ) <-> ( x e. { A } /\ y = B ) )
4	3	opabbii 4717	. 2 |- { <. x , y >. | ( x = A /\ y = B ) } = { <. x , y >. | ( x e. { A } /\ y = B ) }
5		velsn 4193	. . . . 5 |- ( p e. { <. A , C >. } <-> p = <. A , C >. )
6		eqidd 2623	. . . . . . 7 |- ( ( A e. V /\ C e. W ) -> A = A )
7		eqidd 2623	. . . . . . 7 |- ( ( A e. V /\ C e. W ) -> C = C )
8		eqeq1 2626	. . . . . . . . . 10 |- ( x = A -> ( x = A <-> A = A ) )
9	8	adantr 481	. . . . . . . . 9 |- ( ( x = A /\ y = C ) -> ( x = A <-> A = A ) )
10		eqeq1 2626	. . . . . . . . . 10 |- ( y = C -> ( y = B <-> C = B ) )
11		fmptsng.1	. . . . . . . . . . 11 |- ( x = A -> B = C )
12	11	eqeq2d 2632	. . . . . . . . . 10 |- ( x = A -> ( C = B <-> C = C ) )
13	10, 12	sylan9bbr 737	. . . . . . . . 9 |- ( ( x = A /\ y = C ) -> ( y = B <-> C = C ) )
14	9, 13	anbi12d 747	. . . . . . . 8 |- ( ( x = A /\ y = C ) -> ( ( x = A /\ y = B ) <-> ( A = A /\ C = C ) ) )
15	14	opelopabga 4988	. . . . . . 7 |- ( ( A e. V /\ C e. W ) -> ( <. A , C >. e. { <. x , y >. | ( x = A /\ y = B ) } <-> ( A = A /\ C = C ) ) )
16	6, 7, 15	mpbir2and 957	. . . . . 6 |- ( ( A e. V /\ C e. W ) -> <. A , C >. e. { <. x , y >. | ( x = A /\ y = B ) } )
17		eleq1 2689	. . . . . 6 |- ( p = <. A , C >. -> ( p e. { <. x , y >. | ( x = A /\ y = B ) } <-> <. A , C >. e. { <. x , y >. | ( x = A /\ y = B ) } ) )
18	16, 17	syl5ibrcom 237	. . . . 5 |- ( ( A e. V /\ C e. W ) -> ( p = <. A , C >. -> p e. { <. x , y >. | ( x = A /\ y = B ) } ) )
19	5, 18	syl5bi 232	. . . 4 |- ( ( A e. V /\ C e. W ) -> ( p e. { <. A , C >. } -> p e. { <. x , y >. | ( x = A /\ y = B ) } ) )
20		elopab 4983	. . . . 5 |- ( p e. { <. x , y >. | ( x = A /\ y = B ) } <-> E. x E. y ( p = <. x , y >. /\ ( x = A /\ y = B ) ) )
21		opeq12 4404	. . . . . . . . . 10 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
22	21	eqeq2d 2632	. . . . . . . . 9 |- ( ( x = A /\ y = B ) -> ( p = <. x , y >. <-> p = <. A , B >. ) )
23	11	adantr 481	. . . . . . . . . . . 12 |- ( ( x = A /\ y = B ) -> B = C )
24	23	opeq2d 4409	. . . . . . . . . . 11 |- ( ( x = A /\ y = B ) -> <. A , B >. = <. A , C >. )
25		opex 4932	. . . . . . . . . . . 12 |- <. A , C >. e. _V
26	25	snid 4208	. . . . . . . . . . 11 |- <. A , C >. e. { <. A , C >. }
27	24, 26	syl6eqel 2709	. . . . . . . . . 10 |- ( ( x = A /\ y = B ) -> <. A , B >. e. { <. A , C >. } )
28		eleq1 2689	. . . . . . . . . 10 |- ( p = <. A , B >. -> ( p e. { <. A , C >. } <-> <. A , B >. e. { <. A , C >. } ) )
29	27, 28	syl5ibrcom 237	. . . . . . . . 9 |- ( ( x = A /\ y = B ) -> ( p = <. A , B >. -> p e. { <. A , C >. } ) )
30	22, 29	sylbid 230	. . . . . . . 8 |- ( ( x = A /\ y = B ) -> ( p = <. x , y >. -> p e. { <. A , C >. } ) )
31	30	impcom 446	. . . . . . 7 |- ( ( p = <. x , y >. /\ ( x = A /\ y = B ) ) -> p e. { <. A , C >. } )
32	31	exlimivv 1860	. . . . . 6 |- ( E. x E. y ( p = <. x , y >. /\ ( x = A /\ y = B ) ) -> p e. { <. A , C >. } )
33	32	a1i 11	. . . . 5 |- ( ( A e. V /\ C e. W ) -> ( E. x E. y ( p = <. x , y >. /\ ( x = A /\ y = B ) ) -> p e. { <. A , C >. } ) )
34	20, 33	syl5bi 232	. . . 4 |- ( ( A e. V /\ C e. W ) -> ( p e. { <. x , y >. | ( x = A /\ y = B ) } -> p e. { <. A , C >. } ) )
35	19, 34	impbid 202	. . 3 |- ( ( A e. V /\ C e. W ) -> ( p e. { <. A , C >. } <-> p e. { <. x , y >. | ( x = A /\ y = B ) } ) )
36	35	eqrdv 2620	. 2 |- ( ( A e. V /\ C e. W ) -> { <. A , C >. } = { <. x , y >. | ( x = A /\ y = B ) } )
37		df-mpt 4730	. . 3 |- ( x e. { A } |-> B ) = { <. x , y >. | ( x e. { A } /\ y = B ) }
38	37	a1i 11	. 2 |- ( ( A e. V /\ C e. W ) -> ( x e. { A } |-> B ) = { <. x , y >. | ( x e. { A } /\ y = B ) } )
39	4, 36, 38	3eqtr4a 2682	1 |- ( ( A e. V /\ C e. W ) -> { <. A , C >. } = ( x e. { A } |-> B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {csn 4177   <.cop 4183   {copab 4712   |-> cmpt 4729

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-opab 4713  df-mpt 4730

This theorem is referenced by:  mdet0pr  20398  m1detdiag  20403