Theorem funopdmsn 5818

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 5338	. . . 4 |- ( Fun G <-> Fun <. X , Y >. )
3		funopdmsn.x	. . . . . 6 |- X e. V
4	3	elexi 2812	. . . . 5 |- X e. _V
5		funopdmsn.y	. . . . . 6 |- Y e. W
6	5	elexi 2812	. . . . 5 |- Y e. _V
7	4, 6	funop 5817	. . . 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 2233	. . . . . . 7 |- <. X , Y >. = G
10	9	eqeq1i 2237	. . . . . 6 |- ( <. X , Y >. = { <. x , x >. } <-> G = { <. x , x >. } )
11		dmeq 4922	. . . . . . . 8 |- ( G = { <. x , x >. } -> dom G = dom { <. x , x >. } )
12		vex 2802	. . . . . . . . 9 |- x e. _V
13	12	dmsnop 5201	. . . . . . . 8 |- dom { <. x , x >. } = { x }
14	11, 13	eqtrdi 2278	. . . . . . 7 |- ( G = { <. x , x >. } -> dom G = { x } )
15		eleq2 2293	. . . . . . . . 9 |- ( dom G = { x } -> ( A e. dom G <-> A e. { x } ) )
16		eleq2 2293	. . . . . . . . 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 3684	. . . . . . . . 9 |- ( A e. { x } -> A = x )
19		elsni 3684	. . . . . . . . 9 |- ( B e. { x } -> B = x )
20		eqtr3 2249	. . . . . . . . 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 1644	. . 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 1225	1 |- ( ( Fun G /\ A e. dom G /\ B e. dom G ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   {csn 3666   <.cop 3669   dom cdm 4718   Fun wfun 5311

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-fun 5319

This theorem is referenced by:  fundm2domnop0  11062