Theorem fsn2g 6405

Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by Thierry Arnoux, 11-Jul-2020.)

Assertion

Ref	Expression
fsn2g	|- ( A e. V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )

Proof of Theorem fsn2g

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4187	. . . 4 |- ( a = A -> { a } = { A } )
2	1	feq2d 6031	. . 3 |- ( a = A -> ( F : { a } --> B <-> F : { A } --> B ) )
3		fveq2 6191	. . . . 5 |- ( a = A -> ( F ` a ) = ( F ` A ) )
4	3	eleq1d 2686	. . . 4 |- ( a = A -> ( ( F ` a ) e. B <-> ( F ` A ) e. B ) )
5		eqidd 2623	. . . . 5 |- ( a = A -> F = F )
6		id 22	. . . . . . 7 |- ( a = A -> a = A )
7	6, 3	opeq12d 4410	. . . . . 6 |- ( a = A -> <. a , ( F ` a ) >. = <. A , ( F ` A ) >. )
8	7	sneqd 4189	. . . . 5 |- ( a = A -> { <. a , ( F ` a ) >. } = { <. A , ( F ` A ) >. } )
9	5, 8	eqeq12d 2637	. . . 4 |- ( a = A -> ( F = { <. a , ( F ` a ) >. } <-> F = { <. A , ( F ` A ) >. } ) )
10	4, 9	anbi12d 747	. . 3 |- ( a = A -> ( ( ( F ` a ) e. B /\ F = { <. a , ( F ` a ) >. } ) <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
11	2, 10	bibi12d 335	. 2 |- ( a = A -> ( ( F : { a } --> B <-> ( ( F ` a ) e. B /\ F = { <. a , ( F ` a ) >. } ) ) <-> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) ) )
12		vex 3203	. . 3 |- a e. _V
13	12	fsn2 6403	. 2 |- ( F : { a } --> B <-> ( ( F ` a ) e. B /\ F = { <. a , ( F ` a ) >. } ) )
14	11, 13	vtoclg 3266	1 |- ( A e. V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   <.cop 4183   -->wf 5884   ` cfv 5888

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-reu 2919  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  fsnex  6538  pt1hmeo  21609  k0004val0  38452  difmapsn  39404