Theorem opbr1st 5501

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

Hypotheses

Ref	Expression
opbr1st.1	⊢ A ∈ V
opbr1st.2	⊢ B ∈ V

Assertion

Ref	Expression
opbr1st	⊢ (⟨A, B⟩1st C ↔ 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 4689	. . 3 ⊢ (⟨A, B⟩1st C → (⟨A, B⟩ ∈ V ∧ C ∈ V))
2	1	simprd 449	. 2 ⊢ (⟨A, B⟩1st C → C ∈ V)
3		opbr1st.1	. . 3 ⊢ A ∈ V
4		eleq1 2413	. . 3 ⊢ (A = C → (A ∈ V ↔ C ∈ V))
5	3, 4	mpbii 202	. 2 ⊢ (A = C → C ∈ V)
6		breq2 4643	. . 3 ⊢ (x = C → (⟨A, B⟩1st x ↔ ⟨A, B⟩1st C))
7		eqeq2 2362	. . 3 ⊢ (x = C → (A = x ↔ A = C))
8		vex 2862	. . . . 5 ⊢ x ∈ V
9	8	br1st 4858	. . . 4 ⊢ (⟨A, B⟩1st x ↔ ∃y⟨A, B⟩ = ⟨x, y⟩)
10		opbr1st.2	. . . . . 6 ⊢ B ∈ V
11		biidd 228	. . . . . 6 ⊢ (y = B → (x = A ↔ x = A))
12	10, 11	ceqsexv 2894	. . . . 5 ⊢ (∃y(y = B ∧ x = A) ↔ x = A)
13		eqcom 2355	. . . . . . 7 ⊢ (⟨A, B⟩ = ⟨x, y⟩ ↔ ⟨x, y⟩ = ⟨A, B⟩)
14		opth 4602	. . . . . . . 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 ⊢ (∃y⟨A, B⟩ = ⟨x, y⟩ ↔ ∃y(y = B ∧ x = A))
19		eqcom 2355	. . . . 5 ⊢ (A = x ↔ x = A)
20	12, 18, 19	3bitr4i 268	. . . 4 ⊢ (∃y⟨A, B⟩ = ⟨x, y⟩ ↔ A = x)
21	9, 20	bitri 240	. . 3 ⊢ (⟨A, B⟩1st x ↔ A = x)
22	6, 7, 21	vtoclbg 2915	. 2 ⊢ (C ∈ V → (⟨A, B⟩1st C ↔ A = C))
23	2, 5, 22	pm5.21nii 342	1 ⊢ (⟨A, B⟩1st C ↔ A = C)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ⟨cop 4561   class class class wbr 4639  1^st c1st 4717

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723

This theorem is referenced by:  1stfo  5505  opfv1st  5514  brco1st  5777  trtxp  5781  op1st2nd  5790  oqelins4  5794  qrpprod  5836  xpassenlem  6056  xpassen  6057  enpw1lem1  6061  nncdiv3lem1  6275