Theorem funsneqopsn 6417

Description: A singleton of an ordered pair is an ordered pair of equal singletons if the components are equal. (Contributed by AV, 24-Sep-2020.)

Hypotheses

Ref	Expression
funsndifnop.a	|- A e. _V
funsndifnop.b	|- B e. _V
funsndifnop.g	|- G = { <. A , B >. }

Assertion

Ref	Expression
funsneqopsn	|- ( A = B -> G = <. { A } , { A } >. )

Proof of Theorem funsneqopsn

Step	Hyp	Ref	Expression
1		opeq2 4403	. . . 4 |- ( A = B -> <. A , A >. = <. A , B >. )
2	1	sneqd 4189	. . 3 |- ( A = B -> { <. A , A >. } = { <. A , B >. } )
3		funsndifnop.g	. . 3 |- G = { <. A , B >. }
4	2, 3	syl6reqr 2675	. 2 |- ( A = B -> G = { <. A , A >. } )
5		eqid 2622	. . . 4 |- A = A
6		eqid 2622	. . . 4 |- { A } = { A }
7	5, 6, 6	3pm3.2i 1239	. . 3 |- ( A = A /\ { A } = { A } /\ { A } = { A } )
8		eqeq1 2626	. . . 4 |- ( G = { <. A , A >. } -> ( G = <. { A } , { A } >. <-> { <. A , A >. } = <. { A } , { A } >. ) )
9		funsndifnop.a	. . . . 5 |- A e. _V
10		snex 4908	. . . . 5 |- { A } e. _V
11	9, 9, 10, 10	snopeqop 4969	. . . 4 |- ( { <. A , A >. } = <. { A } , { A } >. <-> ( A = A /\ { A } = { A } /\ { A } = { A } ) )
12	8, 11	syl6bb 276	. . 3 |- ( G = { <. A , A >. } -> ( G = <. { A } , { A } >. <-> ( A = A /\ { A } = { A } /\ { A } = { A } ) ) )
13	7, 12	mpbiri 248	. 2 |- ( G = { <. A , A >. } -> G = <. { A } , { A } >. )
14	4, 13	syl 17	1 |- ( A = B -> G = <. { A } , { A } >. )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   <.cop 4183

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-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  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

This theorem is referenced by:  funsneqop  6418  vtxvalsnop  25933  iedgvalsnop  25934