Theorem opbr1st 5502

Description: Binary relationship of an ordered pair over 1st. (Contributed by SF, 6-Feb-2015.)

Hypotheses

Ref	Expression
opbr1st.1	|- A e. _V
opbr1st.2	|- B e. _V

Assertion

Ref	Expression
opbr1st	|- (<.A, B>.1stC <-> A = C)

Proof of Theorem opbr1st

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brex 4690	. . 3 |- (<.A, B>.1stC -> (<.A, B>. e. _V /\ C e. _V))
2	1	simprd 449	. 2 |- (<.A, B>.1stC -> C e. _V)
3		opbr1st.1	. . 3 |- A e. _V
4		eleq1 2413	. . 3 |- (A = C -> (A e. _V <-> C e. _V))
5	3, 4	mpbii 202	. 2 |- (A = C -> C e. _V)
6		breq2 4644	. . 3 |- (x = C -> (<.A, B>.1stx <-> <.A, B>.1stC))
7		eqeq2 2362	. . 3 |- (x = C -> (A = x <-> A = C))
8		vex 2863	. . . . 5 |- x e. _V
9	8	br1st 4859	. . . 4 |- (<.A, B>.1stx <-> E.y<.A, B>. = <.x, y>.)
10		opbr1st.2	. . . . . 6 |- B e. _V
11		biidd 228	. . . . . 6 |- (y = B -> (x = A <-> x = A))
12	10, 11	ceqsexv 2895	. . . . 5 |- (E.y(y = B /\ x = A) <-> x = A)
13		eqcom 2355	. . . . . . 7 |- (<.A, B>. = <.x, y>. <-> <.x, y>. = <.A, B>.)
14		opth 4603	. . . . . . . 8 |- (<.x, y>. = <.A, B>. <-> (x = A /\ y = B))
15		ancom 437	. . . . . . . 8 |- ((x = A /\ y = B) <-> (y = B /\ x = A))
16	14, 15	bitri 240	. . . . . . 7 |- (<.x, y>. = <.A, B>. <-> (y = B /\ x = A))
17	13, 16	bitri 240	. . . . . 6 |- (<.A, B>. = <.x, y>. <-> (y = B /\ x = A))
18	17	exbii 1582	. . . . 5 |- (E.y<.A, B>. = <.x, y>. <-> E.y(y = B /\ x = A))
19		eqcom 2355	. . . . 5 |- (A = x <-> x = A)
20	12, 18, 19	3bitr4i 268	. . . 4 |- (E.y<.A, B>. = <.x, y>. <-> A = x)
21	9, 20	bitri 240	. . 3 |- (<.A, B>.1stx <-> A = x)
22	6, 7, 21	vtoclbg 2916	. 2 |- (C e. _V -> (<.A, B>.1stC <-> A = C))
23	2, 5, 22	pm5.21nii 342	1 |- (<.A, B>.1stC <-> A = C)

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  _Vcvv 2860  <.cop 4562   class class class wbr 4640  1stc1st 4718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724

This theorem is referenced by:  1stfo  5506  opfv1st  5515  brco1st  5778  trtxp  5782  op1st2nd  5791  oqelins4  5795  qrpprod  5837  xpassenlem  6057  xpassen  6058  enpw1lem1  6062  nncdiv3lem1  6276