Theorem funopdmsn 5763

Description: The domain of a function which is an ordered pair is a singleton. (Contributed by AV, 15-Nov-2021.) (Avoid depending on this detail.)

Hypotheses

Ref	Expression
funopdmsn.g	|- G = <. X , Y >.
funopdmsn.x	|- X e. V
funopdmsn.y	|- Y e. W

Assertion

Ref	Expression
funopdmsn	|- ( ( Fun G /\ A e. dom G /\ B e. dom G ) -> A = B )

Proof of Theorem funopdmsn

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopdmsn.g	. . . . 5 |- G = <. X , Y >.
2	1	funeqi 5291	. . . 4 |- ( Fun G <-> Fun <. X , Y >. )
3		funopdmsn.x	. . . . . 6 |- X e. V
4	3	elexi 2783	. . . . 5 |- X e. _V
5		funopdmsn.y	. . . . . 6 |- Y e. W
6	5	elexi 2783	. . . . 5 |- Y e. _V
7	4, 6	funop 5762	. . . 4 |- ( Fun <. X , Y >. <-> E. x ( X = { x } /\ <. X , Y >. = { <. x , x >. } ) )
8	2, 7	bitri 184	. . 3 |- ( Fun G <-> E. x ( X = { x } /\ <. X , Y >. = { <. x , x >. } ) )
9	1	eqcomi 2208	. . . . . . 7 |- <. X , Y >. = G
10	9	eqeq1i 2212	. . . . . 6 |- ( <. X , Y >. = { <. x , x >. } <-> G = { <. x , x >. } )
11		dmeq 4877	. . . . . . . 8 |- ( G = { <. x , x >. } -> dom G = dom { <. x , x >. } )
12		vex 2774	. . . . . . . . 9 |- x e. _V
13	12	dmsnop 5155	. . . . . . . 8 |- dom { <. x , x >. } = { x }
14	11, 13	eqtrdi 2253	. . . . . . 7 |- ( G = { <. x , x >. } -> dom G = { x } )
15		eleq2 2268	. . . . . . . . 9 |- ( dom G = { x } -> ( A e. dom G <-> A e. { x } ) )
16		eleq2 2268	. . . . . . . . 9 |- ( dom G = { x } -> ( B e. dom G <-> B e. { x } ) )
17	15, 16	anbi12d 473	. . . . . . . 8 |- ( dom G = { x } -> ( ( A e. dom G /\ B e. dom G ) <-> ( A e. { x } /\ B e. { x } ) ) )
18		elsni 3650	. . . . . . . . 9 |- ( A e. { x } -> A = x )
19		elsni 3650	. . . . . . . . 9 |- ( B e. { x } -> B = x )
20		eqtr3 2224	. . . . . . . . 9 |- ( ( A = x /\ B = x ) -> A = B )
21	18, 19, 20	syl2an 289	. . . . . . . 8 |- ( ( A e. { x } /\ B e. { x } ) -> A = B )
22	17, 21	biimtrdi 163	. . . . . . 7 |- ( dom G = { x } -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
23	14, 22	syl 14	. . . . . 6 |- ( G = { <. x , x >. } -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
24	10, 23	sylbi 121	. . . . 5 |- ( <. X , Y >. = { <. x , x >. } -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
25	24	adantl 277	. . . 4 |- ( ( X = { x } /\ <. X , Y >. = { <. x , x >. } ) -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
26	25	exlimiv 1620	. . 3 |- ( E. x ( X = { x } /\ <. X , Y >. = { <. x , x >. } ) -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
27	8, 26	sylbi 121	. 2 |- ( Fun G -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
28	27	3impib 1203	1 |- ( ( Fun G /\ A e. dom G /\ B e. dom G ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   = wceq 1372   E.wex 1514   e. wcel 2175   {csn 3632   <.cop 3635   dom cdm 4674   Fun wfun 5264

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-fun 5272

This theorem is referenced by:  fundm2domnop0  10988