Theorem funopdmsn 6415

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

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 5909	. . . 4 |- ( Fun G <-> Fun <. X , Y >. )
3		funopdmsn.x	. . . . . 6 |- X e. V
4	3	elexi 3213	. . . . 5 |- X e. _V
5		funopdmsn.y	. . . . . 6 |- Y e. W
6	5	elexi 3213	. . . . 5 |- Y e. _V
7	4, 6	funop 6414	. . . 4 |- ( Fun <. X , Y >. <-> E. x ( X = { x } /\ <. X , Y >. = { <. x , x >. } ) )
8	2, 7	bitri 264	. . 3 |- ( Fun G <-> E. x ( X = { x } /\ <. X , Y >. = { <. x , x >. } ) )
9	1	eqcomi 2631	. . . . . . 7 |- <. X , Y >. = G
10	9	eqeq1i 2627	. . . . . 6 |- ( <. X , Y >. = { <. x , x >. } <-> G = { <. x , x >. } )
11		dmeq 5324	. . . . . . . 8 |- ( G = { <. x , x >. } -> dom G = dom { <. x , x >. } )
12		vex 3203	. . . . . . . . 9 |- x e. _V
13	12	dmsnop 5609	. . . . . . . 8 |- dom { <. x , x >. } = { x }
14	11, 13	syl6eq 2672	. . . . . . 7 |- ( G = { <. x , x >. } -> dom G = { x } )
15		eleq2 2690	. . . . . . . . 9 |- ( dom G = { x } -> ( A e. dom G <-> A e. { x } ) )
16		eleq2 2690	. . . . . . . . 9 |- ( dom G = { x } -> ( B e. dom G <-> B e. { x } ) )
17	15, 16	anbi12d 747	. . . . . . . 8 |- ( dom G = { x } -> ( ( A e. dom G /\ B e. dom G ) <-> ( A e. { x } /\ B e. { x } ) ) )
18		elsni 4194	. . . . . . . . 9 |- ( A e. { x } -> A = x )
19		elsni 4194	. . . . . . . . 9 |- ( B e. { x } -> B = x )
20		eqtr3 2643	. . . . . . . . 9 |- ( ( A = x /\ B = x ) -> A = B )
21	18, 19, 20	syl2an 494	. . . . . . . 8 |- ( ( A e. { x } /\ B e. { x } ) -> A = B )
22	17, 21	syl6bi 243	. . . . . . 7 |- ( dom G = { x } -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
23	14, 22	syl 17	. . . . . 6 |- ( G = { <. x , x >. } -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
24	10, 23	sylbi 207	. . . . 5 |- ( <. X , Y >. = { <. x , x >. } -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
25	24	adantl 482	. . . 4 |- ( ( X = { x } /\ <. X , Y >. = { <. x , x >. } ) -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
26	25	exlimiv 1858	. . 3 |- ( E. x ( X = { x } /\ <. X , Y >. = { <. x , x >. } ) -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
27	8, 26	sylbi 207	. 2 |- ( Fun G -> ( ( A e. dom G /\ B e. dom G ) -> A = B ) )
28	27	3impib 1262	1 |- ( ( Fun G /\ A e. dom G /\ B e. dom G ) -> A = B )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   {csn 4177   <.cop 4183   dom cdm 5114   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:  fundmge2nop0  13274