Theorem fsn2g 5877

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 3719	. . 3 |- ( a = A -> { a } = { A } )
2	1	feq2d 5519	. 2 |- ( a = A -> ( F : { a } --> B <-> F : { A } --> B ) )
3		fveq2 5693	. . . 4 |- ( a = A -> ( F ` a ) = ( F ` A ) )
4	3	eleq1d 2307	. . 3 |- ( a = A -> ( ( F ` a ) e. B <-> ( F ` A ) e. B ) )
5		id 19	. . . . . 6 |- ( a = A -> a = A )
6	5, 3	opeq12d 3910	. . . . 5 |- ( a = A -> <. a , ( F ` a ) >. = <. A , ( F ` A ) >. )
7	6	sneqd 3721	. . . 4 |- ( a = A -> { <. a , ( F ` a ) >. } = { <. A , ( F ` A ) >. } )
8	7	eqeq2d 2250	. . 3 |- ( a = A -> ( F = { <. a , ( F ` a ) >. } <-> F = { <. A , ( F ` A ) >. } ) )
9	4, 8	anbi12d 477	. 2 |- ( a = A -> ( ( ( F ` a ) e. B /\ F = { <. a , ( F ` a ) >. } ) <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )
10		vex 2824	. . 3 |- a e. _V
11	10	fsn2 5876	. 2 |- ( F : { a } --> B <-> ( ( F ` a ) e. B /\ F = { <. a , ( F ` a ) >. } ) )
12	2, 9, 11	vtoclbg 2884	1 |- ( A e. V -> ( F : { A } --> B <-> ( ( F ` A ) e. B /\ F = { <. A , ( F ` A ) >. } ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {csn 3708   <.cop 3711   -->wf 5371   ` cfv 5375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383

This theorem is referenced by:  gzsumsplit0  14128