Theorem funopdmsn 5864

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 5373	. . . 4 |- ( Fun G <-> Fun <. X , Y >. )
3		funopdmsn.x	. . . . . 6 |- X e. V
4	3	elexi 2826	. . . . 5 |- X e. _V
5		funopdmsn.y	. . . . . 6 |- Y e. W
6	5	elexi 2826	. . . . 5 |- Y e. _V
7	4, 6	funop 5861	. . . 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 2236	. . . . . . 7 |- <. X , Y >. = G
10	9	eqeq1i 2240	. . . . . 6 |- ( <. X , Y >. = { <. x , x >. } <-> G = { <. x , x >. } )
11		dmeq 4956	. . . . . . . 8 |- ( G = { <. x , x >. } -> dom G = dom { <. x , x >. } )
12		vex 2816	. . . . . . . . 9 |- x e. _V
13	12	dmsnop 5236	. . . . . . . 8 |- dom { <. x , x >. } = { x }
14	11, 13	eqtrdi 2281	. . . . . . 7 |- ( G = { <. x , x >. } -> dom G = { x } )
15		eleq2 2296	. . . . . . . . 9 |- ( dom G = { x } -> ( A e. dom G <-> A e. { x } ) )
16		eleq2 2296	. . . . . . . . 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 3707	. . . . . . . . 9 |- ( A e. { x } -> A = x )
19		elsni 3707	. . . . . . . . 9 |- ( B e. { x } -> B = x )
20		eqtr3 2252	. . . . . . . . 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 1647	. . 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 1228	1 |- ( ( Fun G /\ A e. dom G /\ B e. dom G ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   {csn 3689   <.cop 3692   dom cdm 4749   Fun wfun 5346

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-fun 5354

This theorem is referenced by:  fundm2domnop0  11220