Theorem fmptsnd 6435

Description: Express a singleton function in maps-to notation. Deduction form of fmptsng 6434. (Contributed by AV, 4-Aug-2019.)

Hypotheses

Ref	Expression
fmptsnd.1	|- ( ( ph /\ x = A ) -> B = C )
fmptsnd.2	|- ( ph -> A e. V )
fmptsnd.3	|- ( ph -> C e. W )

Assertion

Ref	Expression
fmptsnd	|- ( ph -> { <. A , C >. } = ( x e. { A } |-> B ) )

Distinct variable groups:   x, A   x, C   ph, x

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

Proof of Theorem fmptsnd

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	. . . . . . . 8 |- ( ph -> A = A )
7		eqidd 2623	. . . . . . . 8 |- ( ph -> C = C )
8		sbcan 3478	. . . . . . . . . . 11 |- ( [. C / y ]. ( x = A /\ y = B ) <-> ( [. C / y ]. x = A /\ [. C / y ]. y = B ) )
9		fmptsnd.3	. . . . . . . . . . . . 13 |- ( ph -> C e. W )
10		sbcg 3503	. . . . . . . . . . . . 13 |- ( C e. W -> ( [. C / y ]. x = A <-> x = A ) )
11	9, 10	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( [. C / y ]. x = A <-> x = A ) )
12		eqsbc3 3475	. . . . . . . . . . . . 13 |- ( C e. W -> ( [. C / y ]. y = B <-> C = B ) )
13	9, 12	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( [. C / y ]. y = B <-> C = B ) )
14	11, 13	anbi12d 747	. . . . . . . . . . 11 |- ( ph -> ( ( [. C / y ]. x = A /\ [. C / y ]. y = B ) <-> ( x = A /\ C = B ) ) )
15	8, 14	syl5bb 272	. . . . . . . . . 10 |- ( ph -> ( [. C / y ]. ( x = A /\ y = B ) <-> ( x = A /\ C = B ) ) )
16	15	sbcbidv 3490	. . . . . . . . 9 |- ( ph -> ( [. A / x ]. [. C / y ]. ( x = A /\ y = B ) <-> [. A / x ]. ( x = A /\ C = B ) ) )
17		fmptsnd.2	. . . . . . . . . 10 |- ( ph -> A e. V )
18		eqeq1 2626	. . . . . . . . . . . 12 |- ( x = A -> ( x = A <-> A = A ) )
19	18	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ x = A ) -> ( x = A <-> A = A ) )
20		fmptsnd.1	. . . . . . . . . . . 12 |- ( ( ph /\ x = A ) -> B = C )
21	20	eqeq2d 2632	. . . . . . . . . . 11 |- ( ( ph /\ x = A ) -> ( C = B <-> C = C ) )
22	19, 21	anbi12d 747	. . . . . . . . . 10 |- ( ( ph /\ x = A ) -> ( ( x = A /\ C = B ) <-> ( A = A /\ C = C ) ) )
23	17, 22	sbcied 3472	. . . . . . . . 9 |- ( ph -> ( [. A / x ]. ( x = A /\ C = B ) <-> ( A = A /\ C = C ) ) )
24	16, 23	bitrd 268	. . . . . . . 8 |- ( ph -> ( [. A / x ]. [. C / y ]. ( x = A /\ y = B ) <-> ( A = A /\ C = C ) ) )
25	6, 7, 24	mpbir2and 957	. . . . . . 7 |- ( ph -> [. A / x ]. [. C / y ]. ( x = A /\ y = B ) )
26		opelopabsb 4985	. . . . . . 7 |- ( <. A , C >. e. { <. x , y >. | ( x = A /\ y = B ) } <-> [. A / x ]. [. C / y ]. ( x = A /\ y = B ) )
27	25, 26	sylibr 224	. . . . . 6 |- ( ph -> <. A , C >. e. { <. x , y >. | ( x = A /\ y = B ) } )
28		eleq1 2689	. . . . . 6 |- ( p = <. A , C >. -> ( p e. { <. x , y >. | ( x = A /\ y = B ) } <-> <. A , C >. e. { <. x , y >. | ( x = A /\ y = B ) } ) )
29	27, 28	syl5ibrcom 237	. . . . 5 |- ( ph -> ( p = <. A , C >. -> p e. { <. x , y >. | ( x = A /\ y = B ) } ) )
30	5, 29	syl5bi 232	. . . 4 |- ( ph -> ( p e. { <. A , C >. } -> p e. { <. x , y >. | ( x = A /\ y = B ) } ) )
31		elopab 4983	. . . . 5 |- ( p e. { <. x , y >. | ( x = A /\ y = B ) } <-> E. x E. y ( p = <. x , y >. /\ ( x = A /\ y = B ) ) )
32		opeq12 4404	. . . . . . . . . . . 12 |- ( ( x = A /\ y = B ) -> <. x , y >. = <. A , B >. )
33	32	adantl 482	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> <. x , y >. = <. A , B >. )
34	33	eqeq2d 2632	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( p = <. x , y >. <-> p = <. A , B >. ) )
35	20	adantrr 753	. . . . . . . . . . . . 13 |- ( ( ph /\ ( x = A /\ y = B ) ) -> B = C )
36	35	opeq2d 4409	. . . . . . . . . . . 12 |- ( ( ph /\ ( x = A /\ y = B ) ) -> <. A , B >. = <. A , C >. )
37		opex 4932	. . . . . . . . . . . . 13 |- <. A , C >. e. _V
38	37	snid 4208	. . . . . . . . . . . 12 |- <. A , C >. e. { <. A , C >. }
39	36, 38	syl6eqel 2709	. . . . . . . . . . 11 |- ( ( ph /\ ( x = A /\ y = B ) ) -> <. A , B >. e. { <. A , C >. } )
40		eleq1 2689	. . . . . . . . . . 11 |- ( p = <. A , B >. -> ( p e. { <. A , C >. } <-> <. A , B >. e. { <. A , C >. } ) )
41	39, 40	syl5ibrcom 237	. . . . . . . . . 10 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( p = <. A , B >. -> p e. { <. A , C >. } ) )
42	34, 41	sylbid 230	. . . . . . . . 9 |- ( ( ph /\ ( x = A /\ y = B ) ) -> ( p = <. x , y >. -> p e. { <. A , C >. } ) )
43	42	ex 450	. . . . . . . 8 |- ( ph -> ( ( x = A /\ y = B ) -> ( p = <. x , y >. -> p e. { <. A , C >. } ) ) )
44	43	com23 86	. . . . . . 7 |- ( ph -> ( p = <. x , y >. -> ( ( x = A /\ y = B ) -> p e. { <. A , C >. } ) ) )
45	44	impd 447	. . . . . 6 |- ( ph -> ( ( p = <. x , y >. /\ ( x = A /\ y = B ) ) -> p e. { <. A , C >. } ) )
46	45	exlimdvv 1862	. . . . 5 |- ( ph -> ( E. x E. y ( p = <. x , y >. /\ ( x = A /\ y = B ) ) -> p e. { <. A , C >. } ) )
47	31, 46	syl5bi 232	. . . 4 |- ( ph -> ( p e. { <. x , y >. | ( x = A /\ y = B ) } -> p e. { <. A , C >. } ) )
48	30, 47	impbid 202	. . 3 |- ( ph -> ( p e. { <. A , C >. } <-> p e. { <. x , y >. | ( x = A /\ y = B ) } ) )
49	48	eqrdv 2620	. 2 |- ( ph -> { <. A , C >. } = { <. x , y >. | ( x = A /\ y = B ) } )
50		df-mpt 4730	. . 3 |- ( x e. { A } |-> B ) = { <. x , y >. | ( x e. { A } /\ y = B ) }
51	50	a1i 11	. 2 |- ( ph -> ( x e. { A } |-> B ) = { <. x , y >. | ( x e. { A } /\ y = B ) } )
52	4, 49, 51	3eqtr4a 2682	1 |- ( ph -> { <. A , C >. } = ( x e. { A } |-> B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   [.wsbc 3435   {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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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:  fmptapd  6437  fmptpr  6438  mpt2sn  7268