Theorem br2ndeq 27720

Description: Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)

Hypotheses

Ref	Expression
br2ndeq.1	|- A e. _V
br2ndeq.2	|- B e. _V
br2ndeq.3	|- C e. _V

Assertion

Ref	Expression
br2ndeq	|- ( <. A , B >. 2nd C <-> C = B )

Proof of Theorem br2ndeq

Step	Hyp	Ref	Expression
1		br2ndeq.1	. . . 4 |- A e. _V
2		br2ndeq.2	. . . 4 |- B e. _V
3	1, 2	op2nd 6686	. . 3 |- ( 2nd ` <. A , B >. ) = B
4	3	eqeq1i 2458	. 2 |- ( ( 2nd ` <. A , B >. ) = C <-> B = C )
5		fo2nd 6697	. . . 4 |- 2nd : _V -onto-> _V
6		fofn 5720	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
7	5, 6	ax-mp 5	. . 3 |- 2nd Fn _V
8		opex 4654	. . 3 |- <. A , B >. e. _V
9		fnbrfvb 5831	. . 3 |- ( ( 2nd Fn _V /\ <. A , B >. e. _V ) -> ( ( 2nd ` <. A , B >. ) = C <-> <. A , B >. 2nd C ) )
10	7, 8, 9	mp2an 672	. 2 |- ( ( 2nd ` <. A , B >. ) = C <-> <. A , B >. 2nd C )
11		eqcom 2460	. 2 |- ( B = C <-> C = B )
12	4, 10, 11	3bitr3i 275	1 |- ( <. A , B >. 2nd C <-> C = B )

Syntax hints:   <-> wb 184   = wceq 1370   e. wcel 1758   _Vcvv 3068   <.cop 3981   class class class wbr 4390   Fn wfn 5511   -onto->wfo 5514   ` cfv 5516   2ndc2nd 6676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fo 5522  df-fv 5524  df-2nd 6678

This theorem is referenced by:  dfrn5  27722  brtxp  28045  brpprod  28050  elfuns  28080  brimg  28102  brcup  28104  brcap  28105  brrestrict  28114