Theorem br2ndeq 29370

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 6708	. . 3 |- ( 2nd ` <. A , B >. ) = B
4	3	eqeq1i 2389	. 2 |- ( ( 2nd ` <. A , B >. ) = C <-> B = C )
5		fo2nd 6720	. . . 4 |- 2nd : _V -onto-> _V
6		fofn 5705	. . . 4 |- ( 2nd : _V -onto-> _V -> 2nd Fn _V )
7	5, 6	ax-mp 5	. . 3 |- 2nd Fn _V
8		opex 4626	. . 3 |- <. A , B >. e. _V
9		fnbrfvb 5814	. . 3 |- ( ( 2nd Fn _V /\ <. A , B >. e. _V ) -> ( ( 2nd ` <. A , B >. ) = C <-> <. A , B >. 2nd C ) )
10	7, 8, 9	mp2an 670	. 2 |- ( ( 2nd ` <. A , B >. ) = C <-> <. A , B >. 2nd C )
11		eqcom 2391	. 2 |- ( B = C <-> C = B )
12	4, 10, 11	3bitr3i 275	1 |- ( <. A , B >. 2nd C <-> C = B )

Syntax hints:   <-> wb 184   = wceq 1399   e. wcel 1826   _Vcvv 3034   <.cop 3950   class class class wbr 4367   Fn wfn 5491   -onto->wfo 5494   ` cfv 5496   2ndc2nd 6698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504  df-2nd 6700

This theorem is referenced by:  dfrn5  29372  brtxp  29683  brpprod  29688  elfuns  29718  brimg  29740  brcup  29742  brcap  29743  brrestrict  29752