Theorem br1st 4858

Description: Binary relationship equivalence for the 1^st function. (Contributed by set.mm contributors, 8-Jan-2015.)

Hypothesis

Ref	Expression
br1st.1	⊢ B ∈ V

Assertion

Ref	Expression
br1st	⊢ (A1st B ↔ ∃x A = ⟨B, x⟩)

Distinct variable groups:   x,A   x,B

Proof of Theorem br1st

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

Step	Hyp	Ref	Expression
1		brex 4689	. . 3 ⊢ (A1st B → (A ∈ V ∧ B ∈ V))
2	1	simpld 445	. 2 ⊢ (A1st B → A ∈ V)
3		br1st.1	. . . . 5 ⊢ B ∈ V
4		vex 2862	. . . . 5 ⊢ x ∈ V
5	3, 4	opex 4588	. . . 4 ⊢ ⟨B, x⟩ ∈ V
6		eleq1 2413	. . . 4 ⊢ (A = ⟨B, x⟩ → (A ∈ V ↔ ⟨B, x⟩ ∈ V))
7	5, 6	mpbiri 224	. . 3 ⊢ (A = ⟨B, x⟩ → A ∈ V)
8	7	exlimiv 1634	. 2 ⊢ (∃x A = ⟨B, x⟩ → A ∈ V)
9		eqeq1 2359	. . . . 5 ⊢ (y = A → (y = ⟨z, x⟩ ↔ A = ⟨z, x⟩))
10	9	exbidv 1626	. . . 4 ⊢ (y = A → (∃x y = ⟨z, x⟩ ↔ ∃x A = ⟨z, x⟩))
11		opeq1 4578	. . . . . 6 ⊢ (z = B → ⟨z, x⟩ = ⟨B, x⟩)
12	11	eqeq2d 2364	. . . . 5 ⊢ (z = B → (A = ⟨z, x⟩ ↔ A = ⟨B, x⟩))
13	12	exbidv 1626	. . . 4 ⊢ (z = B → (∃x A = ⟨z, x⟩ ↔ ∃x A = ⟨B, x⟩))
14		df-1st 4723	. . . 4 ⊢ 1st = {⟨y, z⟩ ∣ ∃x y = ⟨z, x⟩}
15	10, 13, 14	brabg 4706	. . 3 ⊢ ((A ∈ V ∧ B ∈ V) → (A1st B ↔ ∃x A = ⟨B, x⟩))
16	3, 15	mpan2 652	. 2 ⊢ (A ∈ V → (A1st B ↔ ∃x A = ⟨B, x⟩))
17	2, 8, 16	pm5.21nii 342	1 ⊢ (A1st B ↔ ∃x A = ⟨B, x⟩)

Syntax hints:   ↔ wb 176  ∃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:  df2nd2  5111  dfxp2  5113  opbr1st  5501  1stfo  5505  dfdm4  5507  brtxp  5783  op1st2nd  5790  addcfnex  5824  brpprod  5839  fundmen  6043  csucex  6259  addccan2nclem1  6263