Theorem funopsn 6413

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

Hypotheses

Ref	Expression
funopsn.x	|- X e. _V
funopsn.y	|- Y e. _V

Assertion

Ref	Expression
funopsn	|- ( ( Fun F /\ F = <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )

Distinct variable groups:   F, a   X, a   Y, a

Proof of Theorem funopsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funiun 6412	. . 3 |- ( Fun F -> F = U_ x e. dom F { <. x , ( F ` x ) >. } )
2		eqeq1 2626	. . . . . . . . . 10 |- ( F = <. X , Y >. -> ( F = U_ x e. dom F { <. x , ( F ` x ) >. } <-> <. X , Y >. = U_ x e. dom F { <. x , ( F ` x ) >. } ) )
3		eqcom 2629	. . . . . . . . . 10 |- ( <. X , Y >. = U_ x e. dom F { <. x , ( F ` x ) >. } <-> U_ x e. dom F { <. x , ( F ` x ) >. } = <. X , Y >. )
4	2, 3	syl6bb 276	. . . . . . . . 9 |- ( F = <. X , Y >. -> ( F = U_ x e. dom F { <. x , ( F ` x ) >. } <-> U_ x e. dom F { <. x , ( F ` x ) >. } = <. X , Y >. ) )
5	4	adantl 482	. . . . . . . 8 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( F = U_ x e. dom F { <. x , ( F ` x ) >. } <-> U_ x e. dom F { <. x , ( F ` x ) >. } = <. X , Y >. ) )
6		funopsn.x	. . . . . . . . . . 11 |- X e. _V
7		funopsn.y	. . . . . . . . . . 11 |- Y e. _V
8	6, 7	opnzi 4943	. . . . . . . . . 10 |- <. X , Y >. =/= (/)
9		neeq1 2856	. . . . . . . . . . . . . 14 |- ( <. X , Y >. = F -> ( <. X , Y >. =/= (/) <-> F =/= (/) ) )
10	9	eqcoms 2630	. . . . . . . . . . . . 13 |- ( F = <. X , Y >. -> ( <. X , Y >. =/= (/) <-> F =/= (/) ) )
11		funrel 5905	. . . . . . . . . . . . . . . . 17 |- ( Fun F -> Rel F )
12		reldm0 5343	. . . . . . . . . . . . . . . . 17 |- ( Rel F -> ( F = (/) <-> dom F = (/) ) )
13	11, 12	syl 17	. . . . . . . . . . . . . . . 16 |- ( Fun F -> ( F = (/) <-> dom F = (/) ) )
14	13	biimprd 238	. . . . . . . . . . . . . . 15 |- ( Fun F -> ( dom F = (/) -> F = (/) ) )
15	14	necon3d 2815	. . . . . . . . . . . . . 14 |- ( Fun F -> ( F =/= (/) -> dom F =/= (/) ) )
16	15	com12 32	. . . . . . . . . . . . 13 |- ( F =/= (/) -> ( Fun F -> dom F =/= (/) ) )
17	10, 16	syl6bi 243	. . . . . . . . . . . 12 |- ( F = <. X , Y >. -> ( <. X , Y >. =/= (/) -> ( Fun F -> dom F =/= (/) ) ) )
18	17	com3l 89	. . . . . . . . . . 11 |- ( <. X , Y >. =/= (/) -> ( Fun F -> ( F = <. X , Y >. -> dom F =/= (/) ) ) )
19	18	impd 447	. . . . . . . . . 10 |- ( <. X , Y >. =/= (/) -> ( ( Fun F /\ F = <. X , Y >. ) -> dom F =/= (/) ) )
20	8, 19	ax-mp 5	. . . . . . . . 9 |- ( ( Fun F /\ F = <. X , Y >. ) -> dom F =/= (/) )
21		fvex 6201	. . . . . . . . . 10 |- ( F ` x ) e. _V
22	21, 6, 7	iunopeqop 4981	. . . . . . . . 9 |- ( dom F =/= (/) -> ( U_ x e. dom F { <. x , ( F ` x ) >. } = <. X , Y >. -> E. a dom F = { a } ) )
23	20, 22	syl 17	. . . . . . . 8 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( U_ x e. dom F { <. x , ( F ` x ) >. } = <. X , Y >. -> E. a dom F = { a } ) )
24	5, 23	sylbid 230	. . . . . . 7 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( F = U_ x e. dom F { <. x , ( F ` x ) >. } -> E. a dom F = { a } ) )
25	24	imp 445	. . . . . 6 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ F = U_ x e. dom F { <. x , ( F ` x ) >. } ) -> E. a dom F = { a } )
26		iuneq1 4534	. . . . . . . . . . . 12 |- ( dom F = { a } -> U_ x e. dom F { <. x , ( F ` x ) >. } = U_ x e. { a } { <. x , ( F ` x ) >. } )
27		vex 3203	. . . . . . . . . . . . 13 |- a e. _V
28		id 22	. . . . . . . . . . . . . . 15 |- ( x = a -> x = a )
29		fveq2 6191	. . . . . . . . . . . . . . 15 |- ( x = a -> ( F ` x ) = ( F ` a ) )
30	28, 29	opeq12d 4410	. . . . . . . . . . . . . 14 |- ( x = a -> <. x , ( F ` x ) >. = <. a , ( F ` a ) >. )
31	30	sneqd 4189	. . . . . . . . . . . . 13 |- ( x = a -> { <. x , ( F ` x ) >. } = { <. a , ( F ` a ) >. } )
32	27, 31	iunxsn 4603	. . . . . . . . . . . 12 |- U_ x e. { a } { <. x , ( F ` x ) >. } = { <. a , ( F ` a ) >. }
33	26, 32	syl6eq 2672	. . . . . . . . . . 11 |- ( dom F = { a } -> U_ x e. dom F { <. x , ( F ` x ) >. } = { <. a , ( F ` a ) >. } )
34	33	adantl 482	. . . . . . . . . 10 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ dom F = { a } ) -> U_ x e. dom F { <. x , ( F ` x ) >. } = { <. a , ( F ` a ) >. } )
35	34	eqeq2d 2632	. . . . . . . . 9 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ dom F = { a } ) -> ( F = U_ x e. dom F { <. x , ( F ` x ) >. } <-> F = { <. a , ( F ` a ) >. } ) )
36		eqeq1 2626	. . . . . . . . . . . . . 14 |- ( F = <. X , Y >. -> ( F = { <. a , ( F ` a ) >. } <-> <. X , Y >. = { <. a , ( F ` a ) >. } ) )
37	36	adantl 482	. . . . . . . . . . . . 13 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( F = { <. a , ( F ` a ) >. } <-> <. X , Y >. = { <. a , ( F ` a ) >. } ) )
38		eqcom 2629	. . . . . . . . . . . . . 14 |- ( <. X , Y >. = { <. a , ( F ` a ) >. } <-> { <. a , ( F ` a ) >. } = <. X , Y >. )
39		fvex 6201	. . . . . . . . . . . . . . 15 |- ( F ` a ) e. _V
40	27, 39, 6, 7	snopeqop 4969	. . . . . . . . . . . . . 14 |- ( { <. a , ( F ` a ) >. } = <. X , Y >. <-> ( a = ( F ` a ) /\ X = Y /\ X = { a } ) )
41	38, 40	sylbb 209	. . . . . . . . . . . . 13 |- ( <. X , Y >. = { <. a , ( F ` a ) >. } -> ( a = ( F ` a ) /\ X = Y /\ X = { a } ) )
42	37, 41	syl6bi 243	. . . . . . . . . . . 12 |- ( ( Fun F /\ F = <. X , Y >. ) -> ( F = { <. a , ( F ` a ) >. } -> ( a = ( F ` a ) /\ X = Y /\ X = { a } ) ) )
43	42	imp 445	. . . . . . . . . . 11 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ F = { <. a , ( F ` a ) >. } ) -> ( a = ( F ` a ) /\ X = Y /\ X = { a } ) )
44		simpr3 1069	. . . . . . . . . . . . . . 15 |- ( ( F = { <. a , ( F ` a ) >. } /\ ( a = ( F ` a ) /\ X = Y /\ X = { a } ) ) -> X = { a } )
45		simp1 1061	. . . . . . . . . . . . . . . . . . . 20 |- ( ( a = ( F ` a ) /\ X = Y /\ X = { a } ) -> a = ( F ` a ) )
46	45	eqcomd 2628	. . . . . . . . . . . . . . . . . . 19 |- ( ( a = ( F ` a ) /\ X = Y /\ X = { a } ) -> ( F ` a ) = a )
47	46	opeq2d 4409	. . . . . . . . . . . . . . . . . 18 |- ( ( a = ( F ` a ) /\ X = Y /\ X = { a } ) -> <. a , ( F ` a ) >. = <. a , a >. )
48	47	sneqd 4189	. . . . . . . . . . . . . . . . 17 |- ( ( a = ( F ` a ) /\ X = Y /\ X = { a } ) -> { <. a , ( F ` a ) >. } = { <. a , a >. } )
49	48	eqeq2d 2632	. . . . . . . . . . . . . . . 16 |- ( ( a = ( F ` a ) /\ X = Y /\ X = { a } ) -> ( F = { <. a , ( F ` a ) >. } <-> F = { <. a , a >. } ) )
50	49	biimpac 503	. . . . . . . . . . . . . . 15 |- ( ( F = { <. a , ( F ` a ) >. } /\ ( a = ( F ` a ) /\ X = Y /\ X = { a } ) ) -> F = { <. a , a >. } )
51	44, 50	jca 554	. . . . . . . . . . . . . 14 |- ( ( F = { <. a , ( F ` a ) >. } /\ ( a = ( F ` a ) /\ X = Y /\ X = { a } ) ) -> ( X = { a } /\ F = { <. a , a >. } ) )
52	51	ex 450	. . . . . . . . . . . . 13 |- ( F = { <. a , ( F ` a ) >. } -> ( ( a = ( F ` a ) /\ X = Y /\ X = { a } ) -> ( X = { a } /\ F = { <. a , a >. } ) ) )
53	52	adantl 482	. . . . . . . . . . . 12 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ F = { <. a , ( F ` a ) >. } ) -> ( ( a = ( F ` a ) /\ X = Y /\ X = { a } ) -> ( X = { a } /\ F = { <. a , a >. } ) ) )
54	53	a1dd 50	. . . . . . . . . . 11 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ F = { <. a , ( F ` a ) >. } ) -> ( ( a = ( F ` a ) /\ X = Y /\ X = { a } ) -> ( dom F = { a } -> ( X = { a } /\ F = { <. a , a >. } ) ) ) )
55	43, 54	mpd 15	. . . . . . . . . 10 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ F = { <. a , ( F ` a ) >. } ) -> ( dom F = { a } -> ( X = { a } /\ F = { <. a , a >. } ) ) )
56	55	impancom 456	. . . . . . . . 9 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ dom F = { a } ) -> ( F = { <. a , ( F ` a ) >. } -> ( X = { a } /\ F = { <. a , a >. } ) ) )
57	35, 56	sylbid 230	. . . . . . . 8 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ dom F = { a } ) -> ( F = U_ x e. dom F { <. x , ( F ` x ) >. } -> ( X = { a } /\ F = { <. a , a >. } ) ) )
58	57	impancom 456	. . . . . . 7 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ F = U_ x e. dom F { <. x , ( F ` x ) >. } ) -> ( dom F = { a } -> ( X = { a } /\ F = { <. a , a >. } ) ) )
59	58	eximdv 1846	. . . . . 6 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ F = U_ x e. dom F { <. x , ( F ` x ) >. } ) -> ( E. a dom F = { a } -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) )
60	25, 59	mpd 15	. . . . 5 |- ( ( ( Fun F /\ F = <. X , Y >. ) /\ F = U_ x e. dom F { <. x , ( F ` x ) >. } ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )
61	60	expcom 451	. . . 4 |- ( F = U_ x e. dom F { <. x , ( F ` x ) >. } -> ( ( Fun F /\ F = <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) )
62	61	expd 452	. . 3 |- ( F = U_ x e. dom F { <. x , ( F ` x ) >. } -> ( Fun F -> ( F = <. X , Y >. -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) ) )
63	1, 62	mpcom 38	. 2 |- ( Fun F -> ( F = <. X , Y >. -> E. a ( X = { a } /\ F = { <. a , a >. } ) ) )
64	63	imp 445	1 |- ( ( Fun F /\ F = <. X , Y >. ) -> E. a ( X = { a } /\ F = { <. a , a >. } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   {csn 4177   <.cop 4183   U_ciun 4520   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   ` 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-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:  funop  6414  funop1  41302