Theorem funop1 41302

Description: A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.)

Assertion

Ref	Expression
funop1	|- ( E. x E. y F = <. x , y >. -> ( Fun F <-> E. x E. y F = { <. x , y >. } ) )

Distinct variable group:   x, F, y

Proof of Theorem funop1

Dummy variables a v w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq12 4404	. . . 4 |- ( ( x = v /\ y = w ) -> <. x , y >. = <. v , w >. )
2	1	eqeq2d 2632	. . 3 |- ( ( x = v /\ y = w ) -> ( F = <. x , y >. <-> F = <. v , w >. ) )
3	2	cbvex2v 2287	. 2 |- ( E. x E. y F = <. x , y >. <-> E. v E. w F = <. v , w >. )
4		vex 3203	. . . . . . 7 |- v e. _V
5		vex 3203	. . . . . . 7 |- w e. _V
6	4, 5	funopsn 6413	. . . . . 6 |- ( ( Fun F /\ F = <. v , w >. ) -> E. a ( v = { a } /\ F = { <. a , a >. } ) )
7		vex 3203	. . . . . . . . 9 |- a e. _V
8		opeq12 4404	. . . . . . . . . . 11 |- ( ( x = a /\ y = a ) -> <. x , y >. = <. a , a >. )
9	8	sneqd 4189	. . . . . . . . . 10 |- ( ( x = a /\ y = a ) -> { <. x , y >. } = { <. a , a >. } )
10	9	eqeq2d 2632	. . . . . . . . 9 |- ( ( x = a /\ y = a ) -> ( F = { <. x , y >. } <-> F = { <. a , a >. } ) )
11	7, 7, 10	spc2ev 3301	. . . . . . . 8 |- ( F = { <. a , a >. } -> E. x E. y F = { <. x , y >. } )
12	11	adantl 482	. . . . . . 7 |- ( ( v = { a } /\ F = { <. a , a >. } ) -> E. x E. y F = { <. x , y >. } )
13	12	exlimiv 1858	. . . . . 6 |- ( E. a ( v = { a } /\ F = { <. a , a >. } ) -> E. x E. y F = { <. x , y >. } )
14	6, 13	syl 17	. . . . 5 |- ( ( Fun F /\ F = <. v , w >. ) -> E. x E. y F = { <. x , y >. } )
15	14	expcom 451	. . . 4 |- ( F = <. v , w >. -> ( Fun F -> E. x E. y F = { <. x , y >. } ) )
16		vex 3203	. . . . . . 7 |- x e. _V
17		vex 3203	. . . . . . 7 |- y e. _V
18	16, 17	funsn 5939	. . . . . 6 |- Fun { <. x , y >. }
19		funeq 5908	. . . . . 6 |- ( F = { <. x , y >. } -> ( Fun F <-> Fun { <. x , y >. } ) )
20	18, 19	mpbiri 248	. . . . 5 |- ( F = { <. x , y >. } -> Fun F )
21	20	exlimivv 1860	. . . 4 |- ( E. x E. y F = { <. x , y >. } -> Fun F )
22	15, 21	impbid1 215	. . 3 |- ( F = <. v , w >. -> ( Fun F <-> E. x E. y F = { <. x , y >. } ) )
23	22	exlimivv 1860	. 2 |- ( E. v E. w F = <. v , w >. -> ( Fun F <-> E. x E. y F = { <. x , y >. } ) )
24	3, 23	sylbi 207	1 |- ( E. x E. y F = <. x , y >. -> ( Fun F <-> E. x E. y F = { <. x , y >. } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   {csn 4177   <.cop 4183   Fun wfun 5882

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-fal 1489  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-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  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: (None)